Recent zbMATH articles in MSC 90Chttps://zbmath.org/atom/cc/90C2023-12-07T16:00:11.105023ZUnknown authorWerkzeugBook review of: S. J. Wright, Primal-dual interior-point methodshttps://zbmath.org/1522.000362023-12-07T16:00:11.105023Z"Cottle, Richard W."https://zbmath.org/authors/?q=ai:cottle.richard-wReview of [Zbl 0863.65031].Book review of: G. Sierksma, Linear and integer programming: Theory and practice. 2nd. ed.https://zbmath.org/1522.000682023-12-07T16:00:11.105023Z"Klamroth, Kathrin"https://zbmath.org/authors/?q=ai:klamroth.kathrinReview of [Zbl 0979.90087].Book review of: E. N. Pistikopoulos et al., Multi-parametric optimization and controlhttps://zbmath.org/1522.000752023-12-07T16:00:11.105023Z"Lavoilles, M."https://zbmath.org/authors/?q=ai:lavoilles.mReview of [Zbl 1471.90001].Book review of: M. Padberg, Linear opimization and extensionshttps://zbmath.org/1522.000882023-12-07T16:00:11.105023Z"Murty, Katta G."https://zbmath.org/authors/?q=ai:murty.katta-gReview of [Zbl 0839.90082].Book review of: M. Parlar, Interactive operations research with Maple. Methods and modelshttps://zbmath.org/1522.000992023-12-07T16:00:11.105023Z"Punnen, Abraham"https://zbmath.org/authors/?q=ai:punnen.abraham-pReview of [Zbl 0967.90001].Book review of: M. H. Veatch, Linear and convex optimization. A mathematical approachhttps://zbmath.org/1522.001062023-12-07T16:00:11.105023Z"Sautto, J. M."https://zbmath.org/authors/?q=ai:sautto.jose-mReview of [Zbl 1505.90002].Book review of: B. S. Mordukhovich and N. M. Nam, Convex analysis and beyond. Volume I. Basic theoryhttps://zbmath.org/1522.001212023-12-07T16:00:11.105023Z"Tammer, Christiane"https://zbmath.org/authors/?q=ai:tammer.christianeReview of [Zbl 1506.90001].Book review of: G. B. Dantzig and M. N. Thapa, Linear programming. 1: Introductionhttps://zbmath.org/1522.001252023-12-07T16:00:11.105023Z"Terlaky, Tamás"https://zbmath.org/authors/?q=ai:terlaky.tamasReview of [Zbl 0883.90090].Book review of: B. S. Mordukhovich and N. M. Nam, Convex analysis and beyond. Volume I. Basic theoryhttps://zbmath.org/1522.001362023-12-07T16:00:11.105023Z"Yen, Nguyen Dong"https://zbmath.org/authors/?q=ai:yen.nguyen-dongReview of [Zbl 1506.90001].Polytops of binary trees, structure of the polytop for the ``snake-type''-treehttps://zbmath.org/1522.050392023-12-07T16:00:11.105023Z"Shcherbakov, Oleg Sergeevich"https://zbmath.org/authors/?q=ai:shcherbakov.oleg-sergeevichSummary: In the paper minimal fillings of finite metric spaces are investigated. This object appeared as a generalization of the concepts of a shortest tree and a minimal filling in the sense of Gromov. It is known that the weight of a minimal filling of a given type can be found by linear programming and by so-called multitours technique. A relation between theses two approaches can be demonstrated using duality in linear programming, namely, rational points of the polytop constructed by the dual problem correspond to multitours. The paper is devoted to investigation of such polytopes, It is shown that the vertices of the polytop are in one-to-one correspondence with irreducible multitours. A description of the polytop and an explicit formula for the weight of the minimal filling of the ``snake-type'' binary tree is obtained.Maximum diameter of 3- and 4-colorable graphshttps://zbmath.org/1522.050802023-12-07T16:00:11.105023Z"Czabarka, Éva"https://zbmath.org/authors/?q=ai:czabarka.eva"Smith, Stephen J."https://zbmath.org/authors/?q=ai:smith.stephen-j"Székely, László"https://zbmath.org/authors/?q=ai:szekely.laszlo-aSummary: \textit{P. Erdős} et al. [J. Comb. Theory, Ser. B 47, No. 1, 73--79 (1989; Zbl 0686.05029)] made conjectures for the maximum diameter of connected graphs without a complete subgraph \(K_{k+1}\), which have order \(n\) and minimum degree \(\delta\). Settling a weaker version of a problem, by strengthening the \(K_{k+1}\)-free condition to \(k\)-colorable, we solve the problem for \(k=3\) and \(k=4\) using a unified linear programming duality approach. The case \(k=4\) is a substantial simplification of the result of
\textit{É. Czabarka} et al. [J. Graph Theory 102, No. 2, 262--270 (2023; Zbl 07746567)].
{{\copyright} 2022 Wiley Periodicals LLC.}Seymour's second-neighborhood conjecture from a different perspectivehttps://zbmath.org/1522.051572023-12-07T16:00:11.105023Z"Bouya, Farid"https://zbmath.org/authors/?q=ai:bouya.farid"Oporowski, Bogdan"https://zbmath.org/authors/?q=ai:oporowski.bogdanSummary: Seymour's Second-Neighborhood Conjecture states that every directed graph whose underlying graph is simple has at least one vertex \(v\) such that the number of vertices of out-distance two from \(v\) is at least as large as the number of vertices of out-distance one from it. We present alternative statements of the conjecture in the language of linear algebra.
{{\copyright} 2021 Wiley Periodicals LLC}The minimal dominating sets in a directed graph and the key indicators set of socio-economic systemhttps://zbmath.org/1522.051672023-12-07T16:00:11.105023Z"Simanchev, Ruslan Yu."https://zbmath.org/authors/?q=ai:simanchev.ruslan-yurevich"Urazova, Inna V."https://zbmath.org/authors/?q=ai:urazova.inna-vladimirovna"Voroshilov, Vladimir V."https://zbmath.org/authors/?q=ai:voroshilov.vladimir-vladimirovichSummary: The paper deals with a digraph with non-negative vertex weights. A subset \(W\) of the set of vertices is called dominating if any vertex that not belongs to it is reachable from the set \(W\) within precisely one step. A dominating set is called minimal if it ceases to be dominating when removing any vertex from it. The paper investigates the problem of searching for a minimal dominating set of maximum weight in a vertex-weighted digraph. An integer linear programming model is proposed for this problem. The model is tested on random instances and the real problem of choosing a family of key indicators in a specific socio-economic system. The paper compares this model with the problem of choosing a dominating set with a fixed number of vertices.Some remarks on hypergraph matching and the Füredi-Kahn-Seymour conjecturehttps://zbmath.org/1522.053362023-12-07T16:00:11.105023Z"Bansal, Nikhil"https://zbmath.org/authors/?q=ai:bansal.nikhil"Harris, David G."https://zbmath.org/authors/?q=ai:harris.david-gSummary: A classic conjecture of \textit{Z. Füredi} et al. [Combinatorica 13, No. 2, 167--180 (1993; Zbl 0779.05030)] states that any hypergraph with non-negative edge weights \(w(e)\) has a matching \(M\) such that \(\sum_{e \in M} (|e|-1+1/|e|) \, w(e) \geq w^{\ast}\), where \(w^{\ast}\) is the value of an optimum fractional matching. We show the conjecture is true for rank-3 hypergraphs and is achieved by a natural iterated rounding algorithm. While the general conjecture remains open, we give several new improved bounds. In particular, we show that the iterated rounding algorithm gives \(\sum_{e \in M} (|e|-\delta (e)) \, w(e) \geq w^{\ast}\), where \(\delta (e)= |e| / (|e|^2+|e|-1)\), improving upon the baseline guarantee of \(\sum_{e \in M} |e| \, w(e) \geq w^{\ast}\).
{{\copyright} 2022 Wiley Periodicals LLC.}Simplex transformations and the multiway cut problemhttps://zbmath.org/1522.054652023-12-07T16:00:11.105023Z"Buchbinder, Niv"https://zbmath.org/authors/?q=ai:buchbinder.niv"Schwartz, Roy"https://zbmath.org/authors/?q=ai:schwartz.roy"Weizman, Baruch"https://zbmath.org/authors/?q=ai:weizman.baruchSummary: We consider multiway cut, a basic graph partitioning problem in which the goal is to find the minimum weight collection of edges disconnecting a given set of special vertices called terminals. Multiway cut admits a well-known simplex embedding relaxation, where rounding this embedding is equivalent to partitioning the simplex. Current best-known solutions to the problem are comprised of a mix of several different ingredients, resulting in intricate algorithms. Moreover, the best of these algorithms is too complex to fully analyze analytically, and a computer was partly used in verifying its approximation factor. We propose a new approach to simplex partitioning and the multiway cut problem based on general transformations of the simplex that allow dependencies between the different variables. Our approach admits much simpler algorithms and, in addition, yields an approximation guarantee for the multiway cut problem that (roughly) matches the current best computer-verified approximation factor.Rational matrix digit systemshttps://zbmath.org/1522.110082023-12-07T16:00:11.105023Z"Jankauskas, Jonas"https://zbmath.org/authors/?q=ai:jankauskas.jonas"Thuswaldner, Jörg M."https://zbmath.org/authors/?q=ai:thuswaldner.jorg-maximilianLet \(A\in\mathrm{GL}_d(\mathbb{Q})\) and define \(\mathbb{Z}^d[A]\) to be the left \(\mathbb{Z}[A]\)-module consisting of all finite sums \(\sum_{i=0}^k A^i\mathbf{x}_i\) with \(k\geq 0\) and \(\mathbf{x}_i\in\mathbb{Z}^d\) for \(i=0,\ldots ,k\). A matrix digit system is a pair \((A,\mathcal{D})\) where \(\mathcal{D}\) is a finite, non-empty subset of \(\mathbb{Z}^d[A]\). Here \(\mathcal{D}\) is called the set of digits of the system. Such a matrix digit system is called finite, if every element of \(\mathbb{Z}^d[A]\) can be expressed as a finite sum \(\sum_{i=0}^k A^i\mathbf{x}_i\), with \(\mathbf{x}_i\in\mathcal{D}\) for \(i=0,\ldots ,k\), i.e., if \(\mathbb{Z}^d[A]=\mathcal{D}[A]\). The problem addressed by the authors is to construct, for a given matrix \(A\in\mathrm{GL}_d(\mathbb{Q})\), a finite set \(\mathcal{D}\) as above such that \((A,\mathcal{D})\) is a finite matrix digit system, and then preferably one for which the set \(\mathcal{D}\) is as small as possible. They give such constructions for various types of matrices. Having constructed such finite digit systems for matrices \(A,B\) they show how to construct one for block matrices \(\Big(\begin{array}{ll} A&0 \\
C&B\end{array}\Big)\). As a result of their investigations, they provide a new, algorithmic, proof of the following theorem that they proved in a previous paper [Linear Algebra Appl. 557, 350--358 (2018; Zbl 1441.11017)]:
\textbf{Theorem.} Let \(A\in\mathrm{GL}_d(\mathbb{Q})\). Then there is \(\mathcal{D}\subset \mathbb{Z}^d[A]\) such that \((A,\mathcal{D})\) is a finite matrix digit system if and only if all eigenvalues of \(A\) in \(\mathbb{C}\) have absolute value \(\geq 1\). In that case, \(\mathcal{D}\) can be chosen as a finite subset of \(\mathbb{Z}^d\).
They illustrate their algorithm to construct such sets \(\mathcal{D}\) with some numerical examples.
Reviewer: Jan-Hendrik Evertse (Leiden)Regularity of powers of Stanley-Reisner ideals of one-dimensional simplicial complexeshttps://zbmath.org/1522.130252023-12-07T16:00:11.105023Z"Minh, Nguyen Cong"https://zbmath.org/authors/?q=ai:nguyen-cong-minh."Vu, Thanh"https://zbmath.org/authors/?q=ai:vu.thanhLet \(\Delta\) be a simplicial complex over the vertex set \(V = \{x_1, \dots, x_n\}\) and let \(I_\Delta\) be its Stanley-Reisner ideal. This paper investigates the Castelnuovo-Mumford regularity of ordinary and symbolic powers of \(I_\Delta\) when \(\dim \Delta = 1\).
When \(\dim \Delta = 1\), the regularity of \(I_\Delta^{(s)}\) was determined by \textit{L. T. Hoa} and \textit{T. N. Trung} [J. Commut. Algebra 8, No. 1, 77--88 (2016; Zbl 1408.13046)]. In this case, the \textit{geometric} regularity of \(I_\Delta^s\) was computed by \textit{D. Lu} [J. Algebr. Comb. 53, No. 4, 991--1014 (2021; Zbl 1469.13038)]. To understand the difference between \(\text{reg }(I_\Delta^{(s)})\) and \(\text{reg }(I_\Delta^s)\), it remains to find the zero-th \(a\)-invariant
\[
a_0(I_\Delta^s) = \max \{q ~\big|~ \left[H^0_{\mathfrak{m}}(I_\Delta^s)\right]_q \not= 0\},
\]
where \({\mathfrak{m}} = (x_1, \dots, x_n)\).
An ideal \(L\) is said to be an \textit{intermediate monomial ideal} between two given monomial ideals \(J \subseteq K\) if \(L = J + (f_1, \dots, f_t)\), where \(f_1, \dots, f_t\) are among the minimal monomial generators of \(K\). The main result of the present paper is stated as follows.
{Theorem 1.1.} Let \(\Delta\) be a one-dimensional simplicial complex. Let \(I = I_\Delta\) be the Stanley-Reisner ideal of \(\Delta\). Then, for all \(s \ge 2\), and all intermediate monomial ideals \(J\) between \(I^s\) and \(I^{(s)}\), we have
\[
\text{reg }(J) = \left\{ \begin{array}{ll} 3x & \text{if } \text{girth } \Delta = 3 \\
2s+1 & \text{if } \text{girth } \Delta = 4 \\
2s & \text{if } \text{girth } \Delta \ge 5. \end{array} \right.
\]
To prove this theorem, the authors examine the \textit{degree complexes} of \(I_\Delta^s\), whose reduced homology groups are directly related to the local cohomology modules of \(I_\Delta^s\). The notion of \textit{degree complexes} was originally introduced by \textit{Y. Takayama} [Bull. Math. Soc. Sci. Math. Roum., Nouv. Sér. 48(96), No. 3, 327--344 (2005; Zbl 1092.13020)] and has seen many interesting applications.
Reviewer: Tai Ha (New Orleans)Exponential convergence of sum-of-squares hierarchies for trigonometric polynomialshttps://zbmath.org/1522.140722023-12-07T16:00:11.105023Z"Bach, Francis"https://zbmath.org/authors/?q=ai:bach.francis-r"Rudi, Alessandro"https://zbmath.org/authors/?q=ai:rudi.alessandroSum-of-Squares (SoS) hierarchies provide a way to compute arbitrary close approximations of a general, global optimization problem over an algebraic or semialgebraic set. In the present paper, the hierarchies of lower bounds for the minimum of trigonometric polynomials on \( [0,1]^n\) and of standard multivariate polynomials on \( [-1,1]^n\) are considered.
First, the authors consider the case of trigonometric polynomials on \([0,1]^n\), without any further assumption. They show that the SoS hierarchy of lower bounds has an error of at most \(O(1/s^2)\), where \(s \in \mathbb{N}\) is the order of the hierarchy. This result translates directly to the SoS hierarchy for standard polynomials on \([-1,1]^n\), which is based on Schmüdgen's Positivstellensatz. In the latter setting, a similar result with error bound \(O(1/s^2)\) was previously obtained by \textit{M. Laurent} and \textit{L. Slot} [Optim. Lett. 17, No. 3, 515--530 (2023; Zbl 1515.90093)].
Second, the paper focuses on trigonometric polynomials on \([0,1]^n\) satisfying generic local optimality conditions. With this extra assumption, a better error bound of at most of the order \(O(\exp (-a \, s)^b)\) is proven. Here, \(a > 0\) and \(b > 1\) are constants depending on \(n\), on the degree \(d\) of the trigonometric polynomial which is minimized, and on local properties of the trigonometric polynomial around its global minimizer. The technique used is based on a quantitative version of truncated Fourier approximation. As in the previous case, this result translates to the SoS hierarchy based on Schmüdgen's Positivstellensatz for standard polynomials on \([-1,1]^n\), which satisfies a local optimality condition.
Under the same local optimality conditions, it was previously shown in [\textit{J. Nie}, Math. Program. 146, No. 1--2 (A), 97--121 (2014; Zbl 1300.65041)] that the SoS hierarchy for standard polynomials has finite convergence. This means that the lower approximation provided by the SoS hierarchy coincides with the true minimum for sufficiently big \(s\). However, since no effective bound is known on \(s\) for this finite convergence to happen, the error bound of the order \(O(\exp (-a \, s)^b)\) remains an important explanation of the practical fast converge rate of the SoS hierarchies.
Reviewer: Lorenzo Baldi (Paris)Spaces that can be ordered effectively: virtually free groups and hyperbolicityhttps://zbmath.org/1522.201802023-12-07T16:00:11.105023Z"Erschler, Anna"https://zbmath.org/authors/?q=ai:erschler.anna"Mitrofanov, Ivan"https://zbmath.org/authors/?q=ai:mitrofanov.ivan-viktorovichLet \((M,d)\), be a metric space and \(X\) a finite subset of \(M\). We denote by \(l_{\mathrm{opt}}(X)\) the minimal length of a path which visits all points of \(X\). Assume that \(T\) is a total order on \(M\). For a finite subset \(X\subset M\), we consider the restriction of order \(T\) on \(M\) and enumerate the points of \(X\) accordingly: \(x_1\leq_{T} x_2\leq_{T} x_3\leq_{T}\dots\leq_{T} x_{k}\) where \(k = \# X\). We denote by \(l_T(X)\) the length of the corresponding path \(l_{T}(X):= d(x_1,x_2) + d(x_2,x_3) + \dots + d(x_{k-1},x_k)\).
Definition 1.1. Given an order space \(M = (M,d,T)\) containing at least two points and \(k\geq 1\), we define the order ratio function
\[
\operatorname{OR}_{M,T}(k):= \sup_{X\subset M\mid 2\leq\# X\leq k+1}\frac{l_{T}(X)}{l_{\mathrm{opt}}(X)}.
\]
If \(M\) consists of a single point, then the supremum in the definition above is taken over an empty set. We use in this case the convention that \(\operatorname{OR}_{M,T}(k) = 1\) for all \(k\geq 1\). For an (unordered) metric space \((M,d)\), we also define the \textit{order ratio function} as \(\operatorname{OR}_{M}(k) := \inf_{T} \operatorname{OR}_{M,T}(k)\). Given an algorithm to find an approximate solution of an optimization problem, the worst case of the ratio between the value provided by this algorithm and the optimal value is called the \textit{competitive ratio.} The traveling salesman problem is a problem to find a cycle of minimum total length that visits each of \(k\) given points. \textit{L. K. Platzman} and \textit{J. J. Bartholdi III} [J. Assoc. Comput. Mach. 36, No. 4, 719--737 (1989; Zbl 0697.68047)] introduced the idea to order all points of a metric space and then, given a \(k\)-point subset, visit its points in the corresponding order. Such an approach is called the \textit{universal travelling salesman problem}. In contrast with previous works on the competitive ratio of universal travelling salesman problem, the authors are interested not only in this asymptotic behaviour but also in particular values of \(\operatorname{OR}(k)\).
Definition 1.2. Let \(M\) be a metric space, containing at least two points, and let \(T\) be an order on \(M\). We say that the order breakpoint \(\operatorname{Br}(M,T) = s\) if \(s\) is the smallest integer such that \(\operatorname{OR}_{M,T}(s) < s\). If such \(s\) does not exists, we say that \(\operatorname{Br}(M,T) = \infty\). In particular, \(\operatorname{Br}(M,T) = 2\) for a one-point space \(M\). Moreover, we define \(\operatorname{Br}(M)\) as the minimum over all orders \(T\) on \(M\). Given an order on \(M\), the order breakpoint describes the minimal value \(k\) for which using this order as a universal order for the travelling salesman problem has some efficiency on \((k+1)\)-point subset. From the definition, \(\operatorname{Br}(M,T)\geq 2\) for any \(M\) and \(\operatorname{Br}(M,T) = 2\) if \(M\) is finite. The order breakpoint is quasi-isometric invariant for uniformly discrete metric spaces, hence it is well-defined for finitelly generated groups. In the paper under review, the following are proved:
Theorem 4.8. Let \(G\) be a finitely generated group. Then \(G\) is virtually free iff \(\operatorname{Br}(G) \leq 3\) (in other words, if \(G\) admits an order \(T\) with \(\operatorname{Br}(G,T) \leq 3\).)
Theorem 5.10. Let \(M\) be a \(\delta\)-hyperbolic graph of bounded degree. Then there exsits an order \(T\) and a constant \(C\) such that for all \(k\geq 1\)
\[
\operatorname{OR}_{M,T}(k)\leq C.
\]
Reviewer: Andrzej Szczepański (Gdańsk)Directional derivatives and subdifferential of convex fuzzy mapping on \(n\)-dimensional time scales and applications to fuzzy programminghttps://zbmath.org/1522.260312023-12-07T16:00:11.105023Z"Wang, Chao"https://zbmath.org/authors/?q=ai:wang.chao"Qin, Guangzhou"https://zbmath.org/authors/?q=ai:qin.guangzhou"Agarwal, Ravi P."https://zbmath.org/authors/?q=ai:agarwal.ravi-pSummary: In this paper, we address the notions of directional derivative, differential and subdifferential of fuzzy mapping \(f : \Lambda^n \to \mathbb{F}^1\), where \(\Lambda^n\) denotes a \(n\)-dimensional time scale and \(\mathbb{F}^1\) is the fuzzy number space. Through using the directional derivative and differential of two crisp functions that are determined by the fuzzy mapping on \(n\)-dimensional time scales, some characterizations of directional derivative and differential are discussed. Moreover, the existence of directional derivative for convex fuzzy mapping is considered on time scales and the relations among directional derivative, differential and subdifferential of fuzzy mapping are established. As applications, some examples of the convex fuzzy programming are given to show the feasibility of our obtained results.Biparametric identification for a free boundary of ductal carcinoma in situhttps://zbmath.org/1522.355162023-12-07T16:00:11.105023Z"Ge, Meibao"https://zbmath.org/authors/?q=ai:ge.meibao"Xu, Dinghua"https://zbmath.org/authors/?q=ai:xu.dinghuaSummary: In this paper we investigate an inverse problem of two parameter identification with free boundary conditions modeling ductal carcinoma in situ (DCIS). Based on the characteristics of the DCIS model, we present an inverse problem of ductal carcinoma in situ (IPDCIS) under the conditions of incisional biopsy measurements at two different moments. Compared with the data in other literatures, this kind of measurements are more feasible and easy to obtain. Moreover, the uniqueness solution to the IPDCIS is proved. The IPDCIS of simultaneously determining unknown parameter and boundary function is transformed into a optimization problem, which can be solved by particle swarm optimization (PSO) method. The numerical simulation results are included to demonstrate the validity of the method and accuracy of the formulation of the IPDCIS. According to the information of clinical incision biopsy, the mathematical model of incision diagnosis of tumor growth pattern is established, and the unknown coefficients in the model are determined based on the proposed mathematical model.Accelerated gradient methods combining Tikhonov regularization with geometric damping driven by the Hessianhttps://zbmath.org/1522.370962023-12-07T16:00:11.105023Z"Attouch, Hedy"https://zbmath.org/authors/?q=ai:attouch.hedy"Balhag, Aïcha"https://zbmath.org/authors/?q=ai:balhag.aicha"Chbani, Zaki"https://zbmath.org/authors/?q=ai:chbani.zaki"Riahi, Hassan"https://zbmath.org/authors/?q=ai:riahi.hassanSummary: In a Hilbert framework, for general convex differentiable optimization, we consider accelerated gradient dynamics combining Tikhonov regularization with Hessian-driven damping. The temporal discretization of these dynamics leads to a new class of first-order optimization algorithms with favorable properties. The Tikhonov regularization parameter is assumed to tend to zero as time tends to infinity, which preserves equilibria. The presence of the Tikhonov regularization term induces a strong property of convexity which vanishes asymptotically. To take advantage of the fast convergence rates attached to the heavy ball method in the strongly convex case, we consider inertial dynamics where the viscous damping coefficient is proportional to the square root of the Tikhonov regularization parameter, and hence converges to zero. The geometric damping, controlled by the Hessian of the function to be minimized, induces attenuation of the oscillations. Under an appropriate setting of the parameters, based on Lyapunov's analysis, we show that the trajectories provide at the same time several remarkable properties: fast convergence of values, fast convergence of gradients towards zero, and strong convergence to the minimum norm minimizer. We show that the corresponding proximal algorithms share the same properties as continuous dynamics. The numerical illustrations confirm the results obtained. This study extends a previous paper by the authors regarding similar problems without the presence of Hessian driven damping.Time rescaling of a primal-dual dynamical system with asymptotically vanishing dampinghttps://zbmath.org/1522.370972023-12-07T16:00:11.105023Z"Hulett, David Alexander"https://zbmath.org/authors/?q=ai:hulett.david-alexander"Nguyen, Dang-Khoa"https://zbmath.org/authors/?q=ai:nguyen.khoa-d|nguyen.dang-khoaSummary: In this work, we approach the minimization of a continuously differentiable convex function under linear equality constraints by a second-order dynamical system with an asymptotically vanishing damping term. The system under consideration is a time rescaled version of another system previously found in the literature. We show fast convergence of the primal-dual gap, the feasibility measure, and the objective function value along the generated trajectories. These convergence rates now depend on the rescaling parameter, and thus can be improved by choosing said parameter appropriately. When the objective function has a Lipschitz continuous gradient, we show that the primal-dual trajectory asymptotically converges weakly to a primal-dual optimal solution to the underlying minimization problem. We also exhibit improved rates of convergence of the gradient along the primal trajectories and of the adjoint of the corresponding linear operator along the dual trajectories. We illustrate the theoretical outcomes and also carry out a comparison with other classes of dynamical systems through numerical experiments.Existence theorems for a variational relation problem of Stampacchia typehttps://zbmath.org/1522.470832023-12-07T16:00:11.105023Z"Balaj, Mircea"https://zbmath.org/authors/?q=ai:balaj.mircea"Serac, Dan Florin"https://zbmath.org/authors/?q=ai:serac.dan-florinSummary: The aim of this paper is to investigate a variational relation problem of Stampacchia type. The existence results presented here are different by the ones obtained in other works which deal with this subject. As applications, we establish existence theorems for two types of variational inequality problems.Existence of solutions and well-posedness for bilevel vector equilibrium problemshttps://zbmath.org/1522.470842023-12-07T16:00:11.105023Z"Chen, Jiawei"https://zbmath.org/authors/?q=ai:chen.jiawei"Ju, Xingxing"https://zbmath.org/authors/?q=ai:ju.xingxing"Liou, Yeong-Cheng"https://zbmath.org/authors/?q=ai:liou.yeongcheng"Wen, Ching-Feng"https://zbmath.org/authors/?q=ai:wen.chingfengSummary: In this paper, we investigate existence of solutions and well-posedness of bilevel vector equilibrium problems. The existence of solution and approximating solution are established by using Fan-KKM theorem. The relations among the uniqueness of solution and the upper semicontinuity or continuity of approximating solution set and the well-posedness of bilevel vector equilibrium problem. We also prove that the generalized well-posedness can be equivalently characterized by the compactness of the solution set and the upper semicontinuity of the approximating solution set. Finally, metric characterizations of the generalized well-posedness are derived in terms of the excess and Kuratowski measure of noncompactness of the approximating solution set under some suitable conditions which ensure the existence of solution of bilevel vector equilibrium problem.An alternated inertial algorithm with weak and linear convergence for solving monotone inclusionshttps://zbmath.org/1522.470872023-12-07T16:00:11.105023Z"Tan, Bing"https://zbmath.org/authors/?q=ai:tan.bing|tan.bing.1"Qin, Xiaolong"https://zbmath.org/authors/?q=ai:qin.xiaolong|qin.xiaolong.1Summary: Inertial-based methods have the drawback of not preserving the Fejér monotonicity of iterative sequences, which may result in slower convergence compared to their corresponding non-inertial versions. To overcome this issue, \textit{Z.-G. Mu} and \textit{Y. Peng} [Stat. Optim. Inf. Comput. 3, No. 3, 241--248 (2015; \url{doi:10.19139/soic.v3i3.124})] suggested an alternating inertial method that can recover the Fejér monotonicity of even subsequences. In this paper, we propose a modified version of the forward-backward algorithm with alternating inertial and relaxation effects to solve an inclusion problem in real Hilbert spaces. The weak and linear convergence of the presented algorithm is established under suitable and mild conditions on the involved operators and parameters. Furthermore, the Fejér monotonicity of even subsequences generated by the proposed algorithm with respect to the solution set is recovered. Finally, our tests on image restoration problems demonstrate the superiority of the proposed algorithm over some related results.A strong convergence algorithm for a fixed point constrained split null point problemhttps://zbmath.org/1522.470972023-12-07T16:00:11.105023Z"Oyewole, O. K."https://zbmath.org/authors/?q=ai:oyewole.olawale-kazeem|oyewole.olalwale-k"Abass, H. A."https://zbmath.org/authors/?q=ai:abass.hammed-anuoluwapo"Mewomo, O. T."https://zbmath.org/authors/?q=ai:mewomo.oluwatosin-temitopeSummary: In this paper, we introduce a new algorithm with self adaptive step-size for finding a common solution of a split feasibility problem and a fixed point problem in real Hilbert spaces. Motivated by the self adaptive step-size method, we incorporate the self adaptive step-size to overcome the difficulty of having to compute the operator norm in the proposed method. Under standard and mild assumption on the control sequences, we establish the strong convergence of the algorithm, obtain a common element in the solution set of a split feasibility problem for sum of two monotone operators and fixed point problem of a demimetric mapping. Numerical examples are presented to illustrate the performance and the behavior of our method. Our result extends, improves and unifies other results in the literature.Refinements of some convergence results of the gradient-projection algorithmhttps://zbmath.org/1522.471002023-12-07T16:00:11.105023Z"Xu, Hong-Kun"https://zbmath.org/authors/?q=ai:xu.hong-kunSummary: In this note we refine some of the results of \textit{H.-K. Xu} [J. Optim. Theory Appl. 150, No. 2, 360--378 (2011; Zbl 1233.90280)] on the gradient-projection algorithm in the infinite-dimensional Hilbert space setting by weakening the conditions imposed on the choices of the parameters in [loc. cit., Theorems 4.2, 4.3 and 5.2]. In addition, we also show that the relaxed gradient-projection algorithm has a sublinear rate of convergence.Resolvent splitting for sums of monotone operators with minimal liftinghttps://zbmath.org/1522.471072023-12-07T16:00:11.105023Z"Malitsky, Yura"https://zbmath.org/authors/?q=ai:malitsky.yura"Tam, Matthew K."https://zbmath.org/authors/?q=ai:tam.matthew-kSummary: In this work, we study fixed point algorithms for finding a zero in the sum of \(n\ge 2\) maximally monotone operators by using their resolvents. More precisely, we consider the class of such algorithms where each resolvent is evaluated only once per iteration. For any algorithm from this class, we show that the underlying fixed point operator is necessarily defined on a \(d\)-fold Cartesian product space with \(d\ge n-1\). Further, we show that this bound is unimprovable by providing a family of examples for which \(d=n-1\) is attained. This family includes the Douglas-Rachford algorithm as the special case when \(n=2\). Applications of the new family of algorithms in distributed decentralised optimisation and multi-block extensions of the alternation direction method of multipliers (ADMM) are discussed.Nonparametric adaptive robust control under model uncertaintyhttps://zbmath.org/1522.490162023-12-07T16:00:11.105023Z"Bayraktar, Erhan"https://zbmath.org/authors/?q=ai:bayraktar.erhan"Chen, Tao"https://zbmath.org/authors/?q=ai:chen.tao.3Summary: We consider a discrete time stochastic Markovian control problem under model uncertainty. Such uncertainty comes not only from the fact that the true probability law of the underlying stochastic process is unknown, but also from the fact that the parametric family of probability distributions to which the true law belongs is unknown. We propose a nonparametric adaptive robust control methodology to deal with such a problem. Our approach hinges on the following building concepts: first, using the adaptive robust paradigm to incorporate online learning and uncertainty reduction into the robust control problem; second, learning the unknown probability law through the empirical distribution, and representing uncertainty reduction in terms of a sequence of Wasserstein balls around the empirical distribution; third, using Lagrangian duality to convert the optimization over Wasserstein balls to a scalar optimization problem, and adopting a machine learning technique to achieve efficient computation of the optimal control. We illustrate our methodology by considering a utility maximization problem. Numerical comparisons show that the nonparametric adaptive robust control approach is preferable to the traditional robust frameworks.Hybrid evolutionary robust optimization-based optimal control for time-delay nonlinear systemshttps://zbmath.org/1522.490192023-12-07T16:00:11.105023Z"Zhang, Jiacheng"https://zbmath.org/authors/?q=ai:zhang.jiacheng"Hou, Ying"https://zbmath.org/authors/?q=ai:hou.ying"Han, Honggui"https://zbmath.org/authors/?q=ai:han.hongguiSummary: Optimal controls receive much attention owing to their remarkable performance for nonlinear systems. However, unknown time-delay disturbances in the optimal control process make it difficult to find high-quality optimal set points. To address this problem, a hybrid evolutionary robust optimization-based optimal control (HERO-OC) method is proposed to reduce the negative effect of time-delay disturbances and enhance the performance of optimal set points. First, a data-driven time-delay disturbance observer (DTDO), based on fuzzy neural networks, is designed to describe the time-delay disturbances of optimal objectives. Then, the expressions and intervals of time-delay disturbances can be estimated to improve the accuracy of optimal objectives. Second, a hybrid evolutionary robust optimization (HERO) algorithm, which combines an expectation robust optimization strategy with a min-max evolutionary robust optimization strategy, is proposed to solve optimal set points. Then, the average robustness and conservative of robustness optimal set points can be improved. Third, an adaptive time-delay controller using Lyapunov-Krasovskii functionals is proposed to track optimal set points. Then, control accuracy is improved while ensuring control stability. Finally, the superior performance of HERO-OC is compared with some novel optimal control methods in a second-order nonlinear system and a wastewater treatment process.On higher-order state constraintshttps://zbmath.org/1522.490252023-12-07T16:00:11.105023Z"Karamzin, Dmitry"https://zbmath.org/authors/?q=ai:karamzin.dmitry-yu"Pereira, Fernando Lobo"https://zbmath.org/authors/?q=ai:lobo-pereira.fernandoThis study deals with higher-order state-constrained optimal control problems and introduces a nondegenerate maximum principle. It relies on unique inward and outward pointing conditions for higher orders, effectively overcoming shortcomings in conventional approaches. Through practical examples, the authors demonstrate how these novel optimality conditions successfully eliminate degenerate Lagrange multiplier sets. Furthermore, the research uncovers a conservation law as a direct outcome of extremality. Lastly, the manuscript investigates the relationship between these innovative conditions and classical optimality criteria, enriching our understanding of optimal control theory.
Reviewer: Suvra Kanti Chakraborty (Kolkata)Well-posedness for bilevel vector equilibrium problems with variable domination structureshttps://zbmath.org/1522.490282023-12-07T16:00:11.105023Z"Xu, Yu-ping"https://zbmath.org/authors/?q=ai:xu.yu-ping"Wang, San-hua"https://zbmath.org/authors/?q=ai:wang.sanhua"Li, Qiu-ying"https://zbmath.org/authors/?q=ai:li.qiuying"Lu, Bing-yi"https://zbmath.org/authors/?q=ai:lu.bing-yiSummary: In this article, well-posedness for two types of bilevel vector equilibrium problems with variable domination structures are introduced and studied. With the help of cosmically upper continuity or Hausdorff upper semi-continuity for variable domination structures, sufficient and necessary conditions are given for such problems to be Levitin-Polyak (LP) well-posed and LP well-posedness in the generalized sense. As variable domination structure is a valid generalization of fixed one, the main results obtained in this article extend and develop some recent works in the literature.Corrigendum to: ``Convex optimization problems on differentiable sets''https://zbmath.org/1522.490292023-12-07T16:00:11.105023Z"Zheng, Xi Yin"https://zbmath.org/authors/?q=ai:zheng.xiyin|zheng.xi-yinSummary: This corrigendum provides a variant of the Brøndsted-Rockafellar theorem, which helps to correct an error in the proof of the sufficiency part of Theorem 2.8 in [\textit{X. Y. Zheng}, ibid. 33, No. 1, 338--359 (2023; Zbl 1510.49021)] and to establish the correct characterization for a closed convex set to be \(C^p\)-differentiable. In particular, in the original paper, Theorem 2.8 holds with \(\varepsilon^{\frac{p}{1+p}}\) replacing \(\varepsilon^{\frac{p}{2}}\), and Theorem 3.2 holds with ``\(\frac{1+p}{p}\)-order-well-posed solvable'' replacing ``\(\frac{2}{p}\)-order-well-posed solvable.'' All other results remain true.Non-Markovian impulse control under nonlinear expectationhttps://zbmath.org/1522.490342023-12-07T16:00:11.105023Z"Perninge, Magnus"https://zbmath.org/authors/?q=ai:perninge.magnusSummary: We consider a general type of non-Markovian impulse control problems under adverse non-linear expectation or, more specifically, the zero-sum game problem where the adversary player decides the probability measure. We show that the upper and lower value functions satisfy a dynamic programming principle (DPP). We first prove the dynamic programming principle (DPP) for a truncated version of the upper value function in a straightforward manner. Relying on a uniform convergence argument then enables us to show the DPP for the general setting. Following this, we use an approximation based on a combination of truncation and discretization to show that the upper and lower value functions coincide, thus establishing that the game has a value and that the DPP holds for the lower value function as well. Finally, we show that the DPP admits a unique solution and give conditions under which a saddle point for the game exists. As an example, we consider a stochastic differential game (SDG) of impulse versus classical control of path-dependent stochastic differential equations (SDEs).A homogeneous Rayleigh quotient with applications in gradient methodshttps://zbmath.org/1522.650492023-12-07T16:00:11.105023Z"Ferrandi, Giulia"https://zbmath.org/authors/?q=ai:ferrandi.giulia"Hochstenbach, Michiel E."https://zbmath.org/authors/?q=ai:hochstenbach.michiel-eSummary: Given an approximate eigenvector, its (standard) Rayleigh quotient and harmonic Rayleigh quotient are two well-known approximations of the corresponding eigenvalue. We propose a new type of Rayleigh quotient, the homogeneous Rayleigh quotient, and analyze its sensitivity with respect to perturbations in the eigenvector. Furthermore, we study the inverse of this homogeneous Rayleigh quotient as stepsize for the gradient method for unconstrained optimization. The notion and basic properties are also extended to the generalized eigenvalue problem.An efficient modified residual-based algorithm for large scale symmetric nonlinear equations by approximating successive iterated gradientshttps://zbmath.org/1522.650782023-12-07T16:00:11.105023Z"Guo, Jie"https://zbmath.org/authors/?q=ai:guo.jie"Wan, Zhong"https://zbmath.org/authors/?q=ai:wan.zhongSummary: In this paper, a new descent approximate modified residual algorithm is developed to solve a large scale system of nonlinear symmetric equations, where the basic strategy to improve its numerical performance is to approximately compute the gradients and the difference of gradients. The error bounds of this approximation are presented, and in virtue of this approximation, a conjugate gradient algorithm for solving large-scale optimization problems in the literature is extended to solve the system of nonlinear symmetric equations without needs of computing and storing the Jacobian matrices or their approximate matrices. It is proved that the obtained search directions in our developed algorithm are sufficiently decent with respect to the so-called approximate modified residues. Under mild assumptions, global and local convergence results of the developed algorithm are proved. Numerical tests indicate that the developed algorithm outperforms the other similar ones available in the literature.A unified convergence analysis of the derivative-free projection-based method for constrained nonlinear monotone equationshttps://zbmath.org/1522.650792023-12-07T16:00:11.105023Z"Ou, Yigui"https://zbmath.org/authors/?q=ai:ou.yigui"Li, Lin"https://zbmath.org/authors/?q=ai:li.lin.2|li.lin.1|li.lin.3|li.lin.6|li.lin.10|li.lin.5|li.lin|li.lin.9Summary: In this paper, we propose a general framework that provides a unified convergence analysis of the derivative-free projection-based method (DFPM) for solving large-scale constrained nonlinear monotone equations. The new results provide a complete picture on the convergence guarantees of DFPM and cover the existing relevant convergence results as special cases. Preliminary numerical experiment results are also reported to show the numerical performance of six line search schemes used in the existing DFPM.A projection-based derivative free DFP approach for solving system of nonlinear convex constrained monotone equations with image restoration applicationshttps://zbmath.org/1522.650802023-12-07T16:00:11.105023Z"ur Rehman, Maaz"https://zbmath.org/authors/?q=ai:ur-rehman.maaz"Sabi'u, Jamilu"https://zbmath.org/authors/?q=ai:sabiu.jamilu"Sohaib, Muhammad"https://zbmath.org/authors/?q=ai:sohaib.muhammad"Shah, Abdullah"https://zbmath.org/authors/?q=ai:shah.abdullahSummary: The nonlinear programming makes use of quasi-Newton methods, a collection of optimization approaches when traditional Newton's method are challenging due to the calculation of the Jacobian matrix and its inverse. Since the Jacobian matrix is computationally difficult to compute and sometimes not available specifically when dealing with non-smooth monotone systems, quasi-Newton methods with superlinear convergence are preferred for solving nonlinear system of equations. This paper provides a new version of the derivative-free David-Fletcher-Powell (DFP) approach for dealing with nonlinear monotone system of equations with convex constraints. The optimal value of the scaling parameter is found by minimizing the condition number of the DFP matrix. Under certain assumptions, the proposed method has global convergence, required minimal storage and is derivative-free. When compared to standard methods, the proposed method requires less iteration, function evaluations, and CPU time. The image restoration test problems demonstrate the method's reliability and efficiency.Generalization of equitable efficiency in multiobjective optimization problems by the direct sum of matriceshttps://zbmath.org/1522.650862023-12-07T16:00:11.105023Z"Ahmadi, F."https://zbmath.org/authors/?q=ai:ahmadi.farshid-farnood|ahmadi.farzane|ahmadi.fatemeh|ahmadi.fatimah-rita|ahmadi.farzad"Salajegheh, A. R."https://zbmath.org/authors/?q=ai:salajegheh.a-r"Foroutannia, D."https://zbmath.org/authors/?q=ai:foroutannia.davoudSummary: We suggest an a priori method by introducing the concept of \(A_P\)-equitable efficiency. The preferences matrix \(A_P\), which is based on the partition \(P\) of the index set of the objective functions, is given by the decision-maker. We state the certain conditions on the matrix \(A_P\) that guarantee the preference relation \(\preceq_{eA_P}\) to satisfy the strict monotonicity and strict \(P\)-transfer principle axioms.
A problem most frequently encountered in multiobjective optimization
is that the set of Pareto optimal solutions provided by the optimization process is a large set. Hence, the decision-making based on selecting a unique preferred solution becomes difficult. Considering models with \(A^r_P\)-equitable efficiency and \(A^\infty_P\)-equitable efficiency can help the decision-maker for overcoming this difficulty, by shrinking the solution set.An asynchronous subgradient-proximal method for solving additive convex optimization problemshttps://zbmath.org/1522.650872023-12-07T16:00:11.105023Z"Arunrat, Tipsuda"https://zbmath.org/authors/?q=ai:arunrat.tipsuda"Namsak, Sakrapee"https://zbmath.org/authors/?q=ai:namsak.sakrapee"Nimana, Nimit"https://zbmath.org/authors/?q=ai:nimana.nimitSummary: In this paper, we consider additive convex optimization problems in which the objective function is the sum of a large number of convex nondifferentiable cost functions. We assume that each cost function is specifically written as the sum of two convex nondifferentiable functions in which one function is appropriate for the subgradient method, and another one is not. To this end, we propose a distributed optimization algorithm based on the subgradient and proximal methods. The proposed method is also governed by an asynchronous feature that allows time-varying delays when computing the subgradients. We prove the convergences of function values of iterates to the optimal value. To demonstrate the efficiency of the presented theoretical result, we investigate the binary classification problem via support vector machine learning.Multi-step inertial strictly contractive PRSM algorithms for convex programming problems with applicationshttps://zbmath.org/1522.650882023-12-07T16:00:11.105023Z"Deng, Zhao"https://zbmath.org/authors/?q=ai:deng.zhao"Han, Deren"https://zbmath.org/authors/?q=ai:han.derenSummary: The Peaceman-Rachford splitting method (PRSM) has been widely studied in recent years because of its excellent numerical performance, especially with inertial acceleration. In this paper, we propose a multi-step inertial strictly contractive PRSM (shortly, MISCPRSM) for solving convex optimization problems, which has not been studied before. This paper aims to give an exhaustive analysis of the value relationship between multiple inertial parameters. The second subproblem utilizes an additional customized indefinite proximal term in order to obtain a closed-form solution. The global convergence and complexity result of the introduced method are analyzed by using variational inequality and basic inequalities. Finally, through simulation and computed tomography experiments, the effectiveness and robustness of the MISCPRSM are proved, and the detailed values of inertial parameters are given.Methodology and first-order algorithms for solving nonsmooth and non-strongly convex bilevel optimization problemshttps://zbmath.org/1522.650892023-12-07T16:00:11.105023Z"Doron, Lior"https://zbmath.org/authors/?q=ai:doron.lior"Shtern, Shimrit"https://zbmath.org/authors/?q=ai:shtern.shimritSummary: Simple bilevel problems are optimization problems in which we want to find an optimal solution to an inner problem that minimizes an outer objective function. Such problems appear in many machine learning and signal processing applications as a way to eliminate undesirable solutions. In our work, we suggest a new approach that is designed for bilevel problems with simple outer functions, such as the \(l_1\) norm, which are not required to be either smooth or strongly convex. In our new ITerative Approximation and Level-set EXpansion (ITALEX) approach, we alternate between expanding the level-set of the outer function and approximately optimizing the inner problem over this level-set. We show that optimizing the inner function through first-order methods such as proximal gradient and generalized conditional gradient results in a feasibility convergence rate of \(O(1/k)\), which up to now was a rate only achieved by bilevel algorithms for smooth and strongly convex outer functions. Moreover, we prove an \(O(1/\sqrt{k})\) rate of convergence for the outer function, contrary to existing methods, which only provide asymptotic guarantees. We demonstrate this performance through numerical experiments.An extended convergence framework applied to complementarity systems with degenerate and nonisolated solutionshttps://zbmath.org/1522.650902023-12-07T16:00:11.105023Z"Fischer, Andreas"https://zbmath.org/authors/?q=ai:fischer.andreas.5"Strasdat, Nico"https://zbmath.org/authors/?q=ai:strasdat.nicoSummary: Some classes of nonlinear complementarity systems, like optimality conditions for generalized Nash equilibrium problems, typically have nonisolated solutions. A reformulation of those systems as a constrained or unconstrained system of equations is often done by means of a nonsmooth complementarity function. Degenerate solutions then lead to points where the reformulated system is nonsmooth. Newton-type methods can have difficulties close to a nonisolated and degenerate solution. For this case, it is known that the LP-Newton method or a constrained Levenberg-Marquardt method may show local superlinear convergence provided that the complemenarity function is piecewise linear. These results rely on error bounds for active pieces of the reformulation. We prove that a related result can be obtained for the Fischer-Burmeister complementarity function on the basis of a somewhat different index error bound condition. To this end, a new convergence framework is developed that allows significantly larger steps. Then, by a sophisticated analysis of the constrained Levenberg-Marquardt method and a corresponding choice of the regularization parameter, local superlinear convergence to a solution with an R-order of 4/3 is shown.To rescale or to project? Solving quadratic programming problems with Lagrange multipliers methodshttps://zbmath.org/1522.650912023-12-07T16:00:11.105023Z"Griva, Igor"https://zbmath.org/authors/?q=ai:griva.igor-aSummary: We conduct numerical experiments to estimate the complexity of two Lagrange multipliers methods for solving quadratic programming (QP) problems with equality constraints and simple bounds. One is based on a fast gradient method treating the bounds with a projection. Another one is based on Newton's method treating the bounds with a nonlinear rescaling principle. Linear equality constraints are handled with an augmented Lagrangian method in both approaches. For the numerical experiments we selected a QP problem for training dual soft margin support vector machines with \(n\) training examples.
We found from the numerical experiments that the time of solving QP problems for training the dual soft-margin SVM grows approximately as \(\mathcal{O}(n^3)\) for both methods, i.e. as a cube of the number of training examples. The results suggest a hypothesis that the time of solving QP problems with inequality constraints may scale up similarly to that of solving QP with just equality constraints, or solving a linear system of equations with \(n\) variables.A Newton-CG based augmented Lagrangian method for finding a second-order stationary point of nonconvex equality constrained optimization with complexity guaranteeshttps://zbmath.org/1522.650922023-12-07T16:00:11.105023Z"He, Chuan"https://zbmath.org/authors/?q=ai:he.chuan"Lu, Zhaosong"https://zbmath.org/authors/?q=ai:lu.zhaosong"Pong, Ting Kei"https://zbmath.org/authors/?q=ai:pong.ting-keiSummary: In this paper we consider finding a second-order stationary point (SOSP) of nonconvex equality constrained optimization when a nearly feasible point is known. In particular, we first propose a new Newton-conjugate gradient (Newton-CG) method for finding an approximate SOSP of unconstrained optimization and show that it enjoys a substantially better complexity than the Newton-CG method in [\textit{C. W. Royer} et al., Math. Program. 180, No. 1--2 (A), 451--488 (2020; Zbl 1448.90081)]. We then propose a Newton-CG based augmented Lagrangian (AL) method for finding an approximate SOSP of nonconvex equality constrained optimization, in which the proposed Newton-CG method is used as a subproblem solver. We show that under a generalized linear independence constraint qualification (GLICQ), our AL method enjoys a total inner iteration complexity of \(\widetilde{\mathcal{O}}(\epsilon^{-7/2})\) and an operation complexity of \(\widetilde{\mathcal{O}}(\epsilon^{-7/2}\min\{n,\epsilon^{-3/4}\})\) for finding an \((\epsilon,\sqrt{\epsilon })\)-SOSP of nonconvex equality constrained optimization with high probability, which are significantly better than the ones achieved by the proximal AL method in [\textit{Y. Xie} and \textit{S. J. Wright}, J. Sci. Comput. 86, No. 3, Paper No. 38, 31 p. (2021; Zbl 1514.90193)]. In addition, we show that it has a total inner iteration complexity of \(\widetilde{\mathcal{O}}(\epsilon^{-11/2})\) and an operation complexity of \(\widetilde{\mathcal{O}}(\epsilon^{-11/2}\min\{n,\epsilon^{-5/4}\})\) when the GLICQ does not hold. To the best of our knowledge, all the complexity results obtained in this paper are new for finding an approximate SOSP of nonconvex equality constrained optimization with high probability. Preliminary numerical results also demonstrate the superiority of our proposed methods over the other competing algorithms.A modified Liu-Storey scheme for nonlinear systems with an application to image recoveryhttps://zbmath.org/1522.650932023-12-07T16:00:11.105023Z"Kiri, A. I."https://zbmath.org/authors/?q=ai:kiri.aliyu-ibrahim"Waziri, M. Y."https://zbmath.org/authors/?q=ai:waziri.mohammed-yusuf"Ahmed, K."https://zbmath.org/authors/?q=ai:ahmed.kabiruSummary: Like the Polak-Ribière-Polyak (PRP) and Hestenes-Stiefel (HS) methods, the classical Liu-Storey (LS) conjugate gradient scheme is widely believed to perform well numerically. This is attributed to the in-built capability of the method to conduct a restart when a bad direction is encountered. However, the scheme's inability to generate descent search directions, which is vital for global convergence, represents its major shortfall. In this article, we present an LS-type scheme for solving system of monotone nonlinear equations with convex constraints. The scheme is based on the approach by \textit{Z. Wang} et al. [Math. Probl. Eng. 2020, Article ID 4381515, 14 p. (2020; Zbl 1459.90169)] and the projection scheme by \textit{M. V. Solodov} and \textit{B. F. Svaiter} [Appl. Optim. 22, 355--369 (1999; Zbl 0928.65059)]. The new scheme satisfies the important condition for global convergence and is suitable for non-smooth nonlinear problems. Furthermore, we demonstrate the method's application in restoring blurry images in compressed sensing. The scheme's global convergence is established under mild assumptions and preliminary numerical results show that the proposed method is promising and performs better than two recent methods in the literature.A class of modulus-based matrix splitting methods for vertical linear complementarity problemhttps://zbmath.org/1522.650942023-12-07T16:00:11.105023Z"Li, Cui-Xia"https://zbmath.org/authors/?q=ai:li.cuixia"Wu, Shi-Liang"https://zbmath.org/authors/?q=ai:wu.shiliangSummary: In this paper, by transforming the vertical linear complementarity problem (VLCP) as a certain absolute value equation, we design a class of modulus-based matrix splitting iteration methods for solving the VLCP. The convergence properties of the proposed methods are discussed in depth. By making use of some numerical experiments, we confirm the efficiency of the proposed methods. Numerical results show that the proposed methods are superior to the classical modulus-based matrix splitting iteration methods.Adaptive piecewise linear relaxations for enclosure computations for nonconvex multiobjective mixed-integer quadratically constrained programshttps://zbmath.org/1522.650952023-12-07T16:00:11.105023Z"Link, Moritz"https://zbmath.org/authors/?q=ai:link.moritz"Volkwein, Stefan"https://zbmath.org/authors/?q=ai:volkwein.stefanSummary: In this paper, a new method for computing an enclosure of the nondominated set of multiobjective mixed-integer quadratically constrained programs without any convexity requirements is presented. In fact, our criterion space method makes use of piecewise linear relaxations in order to bypass the nonconvexity of the original problem. The method chooses adaptively which level of relaxation is needed in which parts of the image space. Furthermore, it is guaranteed that after finitely many iterations, an enclosure of the nondominated set of prescribed quality is returned. We demonstrate the advantages of this approach by applying it to multiobjective energy supply network problems.An efficient hybrid algorithm based on genetic algorithm (GA) and Nelder-Mead (NM) for solving nonlinear inverse parabolic problemshttps://zbmath.org/1522.650962023-12-07T16:00:11.105023Z"Mazraeh, H. Dana"https://zbmath.org/authors/?q=ai:mazraeh.hassan-dana"Pourgholi, R."https://zbmath.org/authors/?q=ai:pourgholi.rezaSummary: In this paper a hybrid algorithm based on genetic algorithm (GA) and Nelder-Mead (NM) simplex search method is combined with least squares method for the determination of temperature in some nonlinear inverse parabolic problems (NIPP). The performance of hybrid algorithm is established with some examples of NIPP. Results show that hybrid algorithm is better than GA and NM separately. Numerical results are obtained by implementation expressed algorithms on 2.20GHz clock speed CPU.Convergence analysis of proximal gradient algorithm with extrapolation for a class of convex nonsmooth minimization problemshttps://zbmath.org/1522.650972023-12-07T16:00:11.105023Z"Pan, Mengxi"https://zbmath.org/authors/?q=ai:pan.mengxi"Wen, Bo"https://zbmath.org/authors/?q=ai:wen.boSummary: We consider the proximal gradient algorithm with extrapolation (PG\(_\mathrm{e}\)) for solving a class of nonsmooth convex minimization problems, whose objective function is the sum of a continuously differentiable convex function with Lipschitz gradient and a proper closed convex function. We first establish the subsequential convergence of iterate generated by PG\(_\mathrm{e}\), then we prove that the convergence rate of objective function is \(O(1/k)\), which implies that the convergence rate of FISTA with fixed restart is also \(O(1/k)\). Finally, we conduct some numerical experiments to illustrate the theoretical results.Relaxation modulus-based matrix splitting iteration method for vertical linear complementarity problemhttps://zbmath.org/1522.650982023-12-07T16:00:11.105023Z"Wang, Dan"https://zbmath.org/authors/?q=ai:wang.dan.1|wang.dan"Li, Jicheng"https://zbmath.org/authors/?q=ai:li.jichengSummary: For solving vertical linear complementarity problems (VLCPs), we propose a relaxation modulus-based matrix splitting iteration method and a two-step relaxation modulus-based matrix splitting iteration method. When the system matrices are \(H_+\)-matrices, we analyze the convergence theory of the proposed methods. Numerical experiments show that relaxation modulus-based matrix splitting iteration method and the two-step relaxation modulus-based matrix splitting iteration method are promising and have great performance in higher dimension.A partially inexact generalized primal-dual hybrid gradient method for saddle point problems with bilinear couplingshttps://zbmath.org/1522.650992023-12-07T16:00:11.105023Z"Wang, Kai"https://zbmath.org/authors/?q=ai:wang.kai.9"Yu, Jintao"https://zbmath.org/authors/?q=ai:yu.jintao"He, Hongjin"https://zbmath.org/authors/?q=ai:he.hongjinSummary: One of the most popular algorithms for saddle point problems is the so-named primal-dual hybrid gradient method, which have been received much considerable attention in the literature. Generally speaking, solving the primal and dual subproblems dominates the main computational cost of those primal-dual type methods. In this paper, we propose a partially inexact generalized primal-dual hybrid gradient method for saddle point problems with bilinear couplings, where the dual subproblem is solved approximately with a relative error strategy. Our proposed algorithm consists of two stages, where the first stage yields a predictor by solving the primal and dual subproblems, and the second procedure makes a correction on the predictor via a simple scheme. It is noteworthy that the underlying extrapolation parameter can be relaxed in a larger range, which allows us to have more choices than a fixed setting. Theoretically, we establish some convergence properties of the proposed algorithm, including the global convergence, the sub-linear convergence rate and the Q-linear convergence rate. Finally, some preliminary computational results demonstrate that our proposed algorithm works well on the fused Lasso problem with synthetic datasets and a pixel-constrained image restoration model.Trace ratio optimization with an application to multi-view learninghttps://zbmath.org/1522.651002023-12-07T16:00:11.105023Z"Wang, Li"https://zbmath.org/authors/?q=ai:wang.li.13"Zhang, Lei-Hong"https://zbmath.org/authors/?q=ai:zhang.lei-hong"Li, Ren-Cang"https://zbmath.org/authors/?q=ai:li.rencangSummary: A trace ratio optimization problem over the Stiefel manifold is investigated from the perspectives of both theory and numerical computations. Necessary conditions in the form of nonlinear eigenvalue problem with eigenvector dependency (NEPv) are established and a numerical method based on the self-consistent field (SCF) iteration with a postprocessing step is designed to solve the NEPv and the method is proved to be always convergent. As an application to multi-view subspace learning, a new framework and its instantiated concrete models are proposed and demonstrated on real world data sets. Numerical results show that the efficiency of the proposed numerical methods and effectiveness of the new orthogonal multi-view subspace learning models.An accelerated descent CG algorithm with clustering the eigenvalues for large-scale nonconvex unconstrained optimization and its application in image restoration problemshttps://zbmath.org/1522.651012023-12-07T16:00:11.105023Z"Wang, Xiaoliang"https://zbmath.org/authors/?q=ai:wang.xiaoliang"Yuan, Gonglin"https://zbmath.org/authors/?q=ai:yuan.gonglinSummary: Conjugate gradient methods are widely used for solving large-scale unconstrained optimization problems. Since they have the attractive practical factors of simple computation and low memory requirement, interesting theoretical features of curvature information and strong global convergence property. Based on the analysis of the minimization of the condition number and the positiveness of the corresponding matrix, we propose a choice for the parameter in Dai-Liao method and design a descent conjugate gradient algorithm which owns the sufficient descent property independent of the choices for line search techniques. Under some common conditions, the global convergence property for uniformly convex function and the general nonlinear function are established. In the numerical experiments, we firstly focus on 46 ill-conditioned matrix problems and present the corresponding primal results. Then 450 large-scale unconstrained problems are referred. Finally, we give an accelerated strategy for the proposed algorithm and apply it to some image restoration problems. Numerical results indicate that the algorithm is reliable and much more efficient and effective than the other methods for the test problems.From optimization to sampling through gradient flowshttps://zbmath.org/1522.651032023-12-07T16:00:11.105023Z"García Trillos, N."https://zbmath.org/authors/?q=ai:garciatrillos.nicolas"Hosseini, B."https://zbmath.org/authors/?q=ai:hosseini.bistoon|hosseini.bamdad|hosseini.barzin|hosseini.boshra|hosseini.bayan|hosseini.babak-sayyid"Sanz-Alonso, D."https://zbmath.org/authors/?q=ai:sanz-alonso.danie|sanz-alonso.danielFrom the text: State-of-the-art algorithms for optimization and sampling often rely on ad-hoc heuristics and empirical tuning, but some unifying principles have emerged that greatly facilitate the understanding of these methods and the communication of algorithmic innovations across scientific communities. This article is concerned with one such principle: the use of gradient flows, and discretizations thereof, to design and analyze optimization and sampling algorithms. The interplay between optimization, sampling, and gradient flows is an active research area and a thorough review of the extant literature is beyond the scope of this article. Our goal is to provide an accessible and lively introduction to some core ideas, emphasizing that gradient flows uncover the conceptual unity behind several existing algorithms and give a rich mathematical framework for their rigorous analysis.Quantum-behaved particle swarm optimization with collaborative attractors for nonlinear numerical problemshttps://zbmath.org/1522.651042023-12-07T16:00:11.105023Z"Liu, Tianyu"https://zbmath.org/authors/?q=ai:liu.tianyu"Jiao, Licheng"https://zbmath.org/authors/?q=ai:jiao.licheng"Ma, Wenping"https://zbmath.org/authors/?q=ai:ma.wenping"Shang, Ronghua"https://zbmath.org/authors/?q=ai:shang.ronghuaSummary: In this paper, an improved quantum-behaved particle swarm optimization (CL-QPSO), which adopts a new collaborative learning strategy to generate local attractors for particles, is proposed to solve nonlinear numerical problems. Local attractors, which directly determine the convergence behavior of particles, play an important role in quantum-behaved particle swarm optimization (QPSO). In order to get a promising and efficient local attractor for each particle, a collaborative learning strategy is introduced to generate local attractors in the proposed algorithm. Collaborative learning strategy consists of two operators, namely orthogonal operator and comparison operator. For each particle, orthogonal operator is used to discover the useful information that lies in its personal and global best positions, while comparison operator is used to enhance the particle's ability of jumping out of local optima. By using a probability parameter, the two operators cooperate with each other to generate local attractors for particles. A comprehensive comparison of CL-QPSO with some state-of-the-art evolutionary algorithms on nonlinear numeric optimization functions demonstrates the effectiveness of the proposed algorithm.Multilevel geometric optimization for regularised constrained linear inverse problemshttps://zbmath.org/1522.651052023-12-07T16:00:11.105023Z"Müller, Sebastian"https://zbmath.org/authors/?q=ai:muller.sebastian.3|muller.sebastian.2|muller.sebastian.4|muller.sebastian.1|muller.sebastian-b"Petra, Stefania"https://zbmath.org/authors/?q=ai:petra.stefania"Zisler, Matthias"https://zbmath.org/authors/?q=ai:zisler.matthiasSummary: We present a geometric multilevel optimization approach that smoothly incorporates box constraints. Given a box constrained optimization problem, we consider a hierarchy of models with varying discretization levels. Finer models are accurate but expensive to compute, while coarser models are less accurate but cheaper to compute. When working at the fine level, multilevel optimization computes the search direction based on a coarser model which speeds up updates at the fine level. Moreover, exploiting geometry induced by the hierarchy the feasibility of the updates is preserved. In particular, our approach extends classical components of multigrid methods like restriction and prolongation to the Riemannian structure of our constraints.Convergence of the RMSProp deep learning method with penalty for nonconvex optimizationhttps://zbmath.org/1522.651092023-12-07T16:00:11.105023Z"Xu, Dongpo"https://zbmath.org/authors/?q=ai:xu.dongpo"Zhang, Shengdong"https://zbmath.org/authors/?q=ai:zhang.shengdong"Zhang, Huisheng"https://zbmath.org/authors/?q=ai:zhang.huisheng"Mandic, Danilo P."https://zbmath.org/authors/?q=ai:mandic.danilo-pSummary: A norm version of the RMSProp algorithm with penalty (termed RMSPropW) is introduced into the deep learning framework and its convergence is addressed both analytically and numerically. For rigour, we consider the general nonconvex setting and prove the boundedness and convergence of the RMSPropW method in both deterministic and stochastic cases. This equips us with strict upper bounds on both the moving average squared norm of the gradient and the norm of weight parameters throughout the learning process, owing to the penalty term within the proposed cost function. In the deterministic (batch) case, the boundedness of the moving average squared norm of the gradient is employed to prove that the gradient sequence converges to zero when using a fixed step size, while with diminishing stepsizes, the minimum of the gradient sequence converges to zero. In the stochastic case, due to the boundedness of the weight evolution sequence, it is further shown that the weight sequence converges to a stationary point with probability 1. Finally, illustrative simulations are provided to support the theoretical analysis, including a comparison with the standard RMSProp on MNIST, CIFAR-10, and IMDB datasets.Approximation, randomization, and combinatorial optimization. Algorithms and techniques, APPROX/RANDOM 2023, Atlanta, Georgia, USA, September 11--13, 2023https://zbmath.org/1522.680382023-12-07T16:00:11.105023ZThe articles of this volume will be reviewed individually. For the preceding conference see [Zbl 1496.68014].Time- and energy-aware task scheduling in environmentally-powered sensor networkshttps://zbmath.org/1522.680772023-12-07T16:00:11.105023Z"Hanschke, Lars"https://zbmath.org/authors/?q=ai:hanschke.lars"Renner, Christian"https://zbmath.org/authors/?q=ai:renner.christianSummary: In the past years, the capabilities and thus application scenarios of Wireless Sensor Networks (WSNs) increased: higher computational power and miniaturization of complex sensors, e.g. fine dust, offer a plethora of new directions. However, energy supply still remains a tough challenge because the use of batteries is neither environmentally-friendly nor maintenance-free. Although energy harvesting promises uninterrupted operation, it requires adaption of the consumption -- which becomes even more complex with increased capabilities of WSNs. In existing literature, adaption to the available energy is typically rate-based. This ignores that the underlying physical phenomena are typically related in time and thus the corresponding sensor tasks cannot be scheduled independently. We close this gap by defining task graphs, allowing arbitrary task relations while including time constraints. To ensure uninterrupted operation of the sensor node, we include energy constraints obtained from a common energy-prediction algorithm. Using a standard Integer Linear Programming (ILP) solver, we generate a schedule for task execution satisfying both time and energy constraints. We exemplarily show, how varying energy resources influence the schedule of a fine dust sensor. Furthermore, we assess the overhead introduced by schedule computation and investigate how the size of the task graph and the available energy affect this overhead. Finally, we present indications for efficiently implementing our approach on sensor nodes.
For the entire collection see [Zbl 1409.68018].Mobility-aware, adaptive algorithms for wireless power transfer in ad hoc networkshttps://zbmath.org/1522.680802023-12-07T16:00:11.105023Z"Madhja, Adelina"https://zbmath.org/authors/?q=ai:madhja.adelina"Nikoletseas, Sotiris"https://zbmath.org/authors/?q=ai:nikoletseas.sotiris-e"Voudouris, Alexandros A."https://zbmath.org/authors/?q=ai:voudouris.alexandros-aSummary: We investigate the interesting impact of mobility on the problem of efficient wireless power transfer in ad hoc networks. We consider a set of mobile agents (consuming energy to perform certain sensing and communication tasks), and a single static charger (with finite energy) which can recharge the agents when they get in its range. In particular, we focus on the problem of efficiently computing the appropriate range of the charger with the goal of prolonging the network lifetime. We first demonstrate (under the realistic assumption of fixed energy supplies) the limitations of any fixed charging range and, therefore, the need for (and power of) a dynamic selection of the charging range, by adapting to the behavior of the mobile agents which is revealed in an online manner. We investigate the complexity of optimizing the selection of such an adaptive charging range, by showing that two simplified offline optimization problems (closely related to the online one) are NP-hard. To effectively address the involved performance trade-offs, we finally present a variety of adaptive heuristics, assuming different levels of agent information regarding their mobility and energy.
For the entire collection see [Zbl 1409.68018].Optimizing probabilities in probabilistic logic programshttps://zbmath.org/1522.680972023-12-07T16:00:11.105023Z"Azzolini, Damiano"https://zbmath.org/authors/?q=ai:azzolini.damiano"Riguzzi, Fabrizio"https://zbmath.org/authors/?q=ai:riguzzi.fabrizioSummary: Probabilistic logic programming is an effective formalism for encoding problems characterized by uncertainty. Some of these problems may require the optimization of probability values subject to constraints among probability distributions of random variables. Here, we introduce a new class of probabilistic logic programs, namely probabilistic \textit{optimizable} logic programs, and we provide an effective algorithm to find the best assignment to probabilities of random variables, such that a set of constraints is satisfied and an objective function is optimized.Elaboration tolerant representation of Markov decision process via decision-theoretic extension of probabilistic action language \(p\mathcal{BC}+\)https://zbmath.org/1522.681162023-12-07T16:00:11.105023Z"Wang, Yi"https://zbmath.org/authors/?q=ai:wang.yi.35"Lee, Joohyung"https://zbmath.org/authors/?q=ai:lee.joohyungSummary: We extend probabilistic action language \(p\mathcal{BC}+\) with the notion of utility in decision theory. The semantics of the extended \(p\mathcal{BC}+\) can be defined as a shorthand notation for a decision-theoretic extension of the probabilistic answer set programming language \(\text{LP}^\text{MLN}\). Alternatively, the semantics of \(p\mathcal{BC}+\) can also be defined in terms of Markov Decision Process (MDP), which in turn allows for representing MDP in a succinct and elaboration tolerant way as well as leveraging an MDP solver to compute a \(p\mathcal{BC}+\) action description. The idea led to the design of the system \textsc{pbcplus2mdp}, which can find an optimal policy of a \(p\mathcal{BC}+\) action description using an MDP solver.
For the entire collection see [Zbl 1410.68009].Elaboration tolerant representation of Markov decision process via decision-theoretic extension of probabilistic action language \(p\mathcal{BC}+\)https://zbmath.org/1522.681172023-12-07T16:00:11.105023Z"Wang, Yi"https://zbmath.org/authors/?q=ai:wang.yi.35"Lee, Joohyung"https://zbmath.org/authors/?q=ai:lee.joohyungSummary: We extend probabilistic action language \(p\mathcal{BC}+\) with the notion of utility in decision theory. The semantics of the extended \(p\mathcal{BC}+\) can be defined as a shorthand notation for a decision-theoretic extension of the probabilistic answer set programming language \(\text{LP}^{\text{MLN}}\). Alternatively, the semantics of \(p\mathcal{BC}+\) can also be defined in terms of Markov decision process (MDP), which in turn allows for representing MDP in a succinct and elaboration tolerant way as well as leveraging an MDP solver to compute a \(p\mathcal{BC}+\) action description. The idea led to the design of the system pbcplus2mdp, which can find an optimal policy of a \(p\mathcal{BC}+\) action description using an MDP solver.Explainable dynamic programminghttps://zbmath.org/1522.681222023-12-07T16:00:11.105023Z"Erwig, Martin"https://zbmath.org/authors/?q=ai:erwig.martin"Kumar, Prashant"https://zbmath.org/authors/?q=ai:kumar.prashant|kumar.prashant.1Summary: In this paper, we present a method for explaining the results produced by dynamic programming (DP) algorithms. Our approach is based on retaining a granular representation of values that are aggregated during program execution. The explanations that are created from the granular representations can answer questions of why one result was obtained instead of another and therefore can increase the confidence in the correctness of program results. Our focus on dynamic programming is motivated by the fact that dynamic programming offers a systematic approach to implementing a large class of optimization algorithms which produce decisions based on aggregated value comparisons. It is those decisions that the granular representation can help explain. Moreover, the fact that dynamic programming can be formalized using semirings supports the creation of a Haskell library for dynamic programming that has two important features. First, it allows programmers to specify programs by recurrence relationships from which efficient implementations are derived automatically. Second, the dynamic programs can be formulated generically (as type classes), which supports the smooth transition from programs that only produce result to programs that can run with granular representation and also produce explanations. Finally, we also demonstrate how to anticipate user questions about program results and how to produce corresponding explanations automatically in advance.Analyzing the quantum annealing approach for solving linear least squares problemshttps://zbmath.org/1522.682312023-12-07T16:00:11.105023Z"Borle, Ajinkya"https://zbmath.org/authors/?q=ai:borle.ajinkya"Lomonaco, Samuel J."https://zbmath.org/authors/?q=ai:lomonaco.samuel-j-junSummary: With the advent of quantum computers, researchers are exploring if quantum mechanics can be leveraged to solve important problems in ways that may provide advantages not possible with conventional or classical methods. A previous work by O'Malley and Vesselinov in 2016 briefly explored using a quantum annealing machine for solving linear least squares problems for real numbers. They suggested that it is best suited for binary and sparse versions of the problem. In our work, we propose a more compact way to represent variables using two's and one's complement on a quantum annealer. We then do an in-depth theoretical analysis of this approach, showing the conditions for which this method may be able to outperform the traditional classical methods for solving general linear least squares problems. Finally, based on our analysis and observations, we discuss potentially promising areas of further research where quantum annealing can be especially beneficial.
For the entire collection see [Zbl 1408.68014].Practical access to dynamic programming on tree decompositionshttps://zbmath.org/1522.682442023-12-07T16:00:11.105023Z"Bannach, Max"https://zbmath.org/authors/?q=ai:bannach.max"Berndt, Sebastian"https://zbmath.org/authors/?q=ai:berndt.sebastianSummary: Parameterized complexity theory has lead to a wide range of algorithmic breakthroughs within the last decades, but the practicability of these methods for real-world problems is still not well understood. We investigate the practicability of one of the fundamental approaches of this field: dynamic programming on tree decompositions. Indisputably, this is a key technique in parameterized algorithms and modern algorithm design. Despite the enormous impact of this approach in theory, it still has very little influence on practical implementations. The reasons for this phenomenon are manifold. One of them is the simple fact that such an implementation requires a long chain of non-trivial tasks (as computing the decomposition, preparing it,\dots). We provide an easy way to implement such dynamic programs that only requires the definition of the update rules. With this interface, dynamic programs for various problems, such as 3-coloring, can be implemented easily in about 100 lines of structured Java code.\par The theoretical foundation of the success of dynamic programming on tree decompositions is well understood due to Courcelle's celebrated theorem, which states that every MSO-definable problem can be efficiently solved if a tree decomposition of small width is given. We seek to provide practical access to this theorem as well, by presenting a lightweight model-checker for a small fragment of MSO. This fragment is powerful enough to describe many natural problems, and our model-checker turns out to be very competitive against similar state-of-the-art tools.
For the entire collection see [Zbl 1393.68010].Schema compliant consistency management via triple graph grammars and integer linear programminghttps://zbmath.org/1522.682572023-12-07T16:00:11.105023Z"Weidmann, Nils"https://zbmath.org/authors/?q=ai:weidmann.nils"Anjorin, Anthony"https://zbmath.org/authors/?q=ai:anjorin.anthonyAlternating good-for-MDPs automatahttps://zbmath.org/1522.682702023-12-07T16:00:11.105023Z"Hahn, Ernst Moritz"https://zbmath.org/authors/?q=ai:hahn.ernst-moritz"Perez, Mateo"https://zbmath.org/authors/?q=ai:perez.mateo"Schewe, Sven"https://zbmath.org/authors/?q=ai:schewe.sven"Somenzi, Fabio"https://zbmath.org/authors/?q=ai:somenzi.fabio"Trivedi, Ashutosh"https://zbmath.org/authors/?q=ai:trivedi.ashutosh"Wojtczak, Dominik"https://zbmath.org/authors/?q=ai:wojtczak.dominikSummary: When omega-regular objectives were first proposed in model-free reinforcement learning (RL) for controlling MDPs, deterministic Rabin automata were used in an attempt to provide a direct translation from their transitions to scalar values. While these translations failed, it has turned out that it is possible to repair them by using good-for-MDPs (GFM) Büchi automata instead. These are nondeterministic Büchi automata with a restricted type of nondeterminism, albeit not as restricted as in good-for-games automata. Indeed, deterministic Rabin automata have a pretty straightforward translation to such GFM automata, which is bi-linear in the number of states and pairs. Interestingly, the same cannot be said for deterministic Streett automata: a translation to nondeterministic Rabin or Büchi automata comes at an exponential cost, even without requiring the target automaton to be good-for-MDPs. Do we have to pay more than that to obtain a good-for-MDPs automaton? The surprising answer is that we have to pay significantly less when we instead expand the good-for-MDPs property to alternating automata: like the nondeterministic GFM automata obtained from deterministic Rabin automata, the alternating good-for-MDPs automata we produce from deterministic Streett automata are bi-linear in the size of the deterministic automaton and its index. They can therefore be exponentially more succinct than the minimal nondeterministic Büchi automaton.
For the entire collection see [Zbl 1511.68011].POMDP controllers with optimal budgethttps://zbmath.org/1522.683462023-12-07T16:00:11.105023Z"Spel, Jip"https://zbmath.org/authors/?q=ai:spel.jip"Stein, Svenja"https://zbmath.org/authors/?q=ai:stein.svenja"Katoen, Joost-Pieter"https://zbmath.org/authors/?q=ai:katoen.joost-pieterSummary: Parametric Markov chains (pMCs) have transitions labeled with functions over a fixed set of parameters. They are useful if the exact transition probabilities are uncertain, e.g., when checking a model for robustness. This paper presents a simple way to check whether the expected total reward until reaching a given target state is monotonic in (some of) the parameters. We exploit this monotonicity together with parameter lifting to find an \(\varepsilon \)-close bound on the optimal expected total reward. Our results are also useful to automatically synthesise controllers with a fixed memory structure for partially observable Markov decision processes (POMDPs), a popular model in AI planning. We experimentally show that our approach can successfully find \(\varepsilon \)-optimal controllers for optimal budget in such POMDPs.
For the entire collection see [Zbl 1511.68006].Active model learning of stochastic reactive systemshttps://zbmath.org/1522.683482023-12-07T16:00:11.105023Z"Tappler, Martin"https://zbmath.org/authors/?q=ai:tappler.martin"Muškardin, Edi"https://zbmath.org/authors/?q=ai:muskardin.edi"Aichernig, Bernhard K."https://zbmath.org/authors/?q=ai:aichernig.bernhard-k"Pill, Ingo"https://zbmath.org/authors/?q=ai:pill.ingoSummary: Black-box systems are inherently hard to verify. Many verification techniques, like model checking, require formal models as a basis. However, such models often do not exist, or they might be outdated. Active automata learning helps to address this issue by offering to automatically infer formal models from system interactions. Hence, automata learning has been receiving much attention in the verification community in recent years. This led to various efficiency improvements, paving the way towards industrial applications. Most research, however, has been focusing on deterministic systems. Here, we present an approach to efficiently learn models of stochastic reactive systems. Our approach adapts \(L^*\)-based learning for Markov decision processes, which we improve and extend to stochastic Mealy machines. Our evaluation demonstrates that we can reduce learning costs by a factor of up to 8.7 in comparison to previous work.
For the entire collection see [Zbl 1511.68014].Dynamic shielding for reinforcement learning in black-box environmentshttps://zbmath.org/1522.683512023-12-07T16:00:11.105023Z"Waga, Masaki"https://zbmath.org/authors/?q=ai:waga.masaki"Castellano, Ezequiel"https://zbmath.org/authors/?q=ai:castellano.ezequiel"Pruekprasert, Sasinee"https://zbmath.org/authors/?q=ai:pruekprasert.sasinee"Klikovits, Stefan"https://zbmath.org/authors/?q=ai:klikovits.stefan"Takisaka, Toru"https://zbmath.org/authors/?q=ai:takisaka.toru"Hasuo, Ichiro"https://zbmath.org/authors/?q=ai:hasuo.ichiroSummary: It is challenging to use reinforcement learning (RL) in cyber-physical systems due to the lack of safety guarantees during learning. Although there have been various proposals to reduce undesired behaviors during learning, most of these techniques require prior system knowledge, and their applicability is limited. This paper aims to reduce undesired behaviors during learning without requiring \textit{any} prior system knowledge. We propose \textit{dynamic shielding}: an extension of a model-based safe RL technique called \textit{shielding} using \textit{automata learning}. The dynamic shielding technique constructs an approximate system model in parallel with RL using a variant of the RPNI algorithm and suppresses undesired explorations due to the shield constructed from the learned model. Through this combination, potentially unsafe actions can be foreseen before the agent experiences them. Experiments show that our dynamic shield significantly decreases the number of undesired events during training.
For the entire collection see [Zbl 1511.68011].Piecewise affine dynamical models of Petri nets -- application to emergency call centershttps://zbmath.org/1522.683562023-12-07T16:00:11.105023Z"Allamigeon, Xavier"https://zbmath.org/authors/?q=ai:allamigeon.xavier"Boyet, Marin"https://zbmath.org/authors/?q=ai:boyet.marin"Gaubert, Stéphane"https://zbmath.org/authors/?q=ai:gaubert.stephaneSummary: We study timed Petri nets, with preselection and priority routing. We represent the behavior of these systems by piecewise affine dynamical systems. We use tools from the theory of nonexpansive mappings to analyze these systems. We establish an equivalence theorem between priority-free fluid timed Petri nets and semi-Markov decision processes, from which we derive the convergence to a periodic regime and the polynomial-time computability of the throughput. More generally, we develop an approach inspired by tropical geometry, characterizing the congestion phases as the cells of a polyhedral complex. We illustrate these results by a current application to the performance evaluation of emergency call centers in the Paris area. We show that priorities can lead to a paradoxical behavior: in certain regimes, the throughput of the most prioritary task may not be an increasing function of the resources.A primal-dual approximation algorithm for \textsc{minsat}https://zbmath.org/1522.683762023-12-07T16:00:11.105023Z"Arif, Umair"https://zbmath.org/authors/?q=ai:arif.umair"Benkoczi, Robert"https://zbmath.org/authors/?q=ai:benkoczi.robert-r"Gaur, Daya Ram"https://zbmath.org/authors/?q=ai:gaur.daya-ram"Krishnamurti, Ramesh"https://zbmath.org/authors/?q=ai:krishnamurti.rameshSummary: We characterize the optimal solution to the linear programming relaxation of the standard formulation for the minimum satisfiability problem. We give a \(O ( n m^2 )\) combinatorial algorithm to solve the fractional version of the minimum satisfiability problem optimally where \(n ( m )\) is the number of variables (clauses). As a by-product, we obtain a \(2 ( 1 - 1 / 2^k )\) approximation algorithm for the minimum satisfiability problem where \(k\) is the maximum number of literals in any clause.Cycles to the rescue! Novel constraints to compute maximum planar subgraphs fasthttps://zbmath.org/1522.683942023-12-07T16:00:11.105023Z"Chimani, Markus"https://zbmath.org/authors/?q=ai:chimani.markus"Wiedera, Tilo"https://zbmath.org/authors/?q=ai:wiedera.tiloSummary: The NP-hard Maximum Planar Subgraph problem asks for a planar subgraph \(H\) of a given graph \(G\) such that \(H\) has maximum edge cardinality. For more than two decades, the only known non-trivial exact algorithm was based on integer linear programming and Kuratowski's famous planarity criterion. We build upon this approach and present new constraint classes -- together with a lifting of the polyhedron -- to obtain provably stronger LP-relaxations, and in turn faster algorithms in practice. The new constraints take Euler's polyhedron formula as a starting point and combine it with considering cycles in \(G\). This paper discusses both the theoretical as well as the practical sides of this strengthening.
For the entire collection see [Zbl 1393.68010].On the maximum connectivity improvement problemhttps://zbmath.org/1522.683972023-12-07T16:00:11.105023Z"Corò, Federico"https://zbmath.org/authors/?q=ai:coro.federico"D'Angelo, Gianlorenzo"https://zbmath.org/authors/?q=ai:dangelo.gianlorenzo"Pinotti, Cristina M."https://zbmath.org/authors/?q=ai:pinotti.cristina-mSummary: In this paper, we define a new problem called the Maximum Connectivity Improvement (MCI) problem: given a directed graph \(G = (V,E)\), a weight function \(w:V \rightarrow \mathbb{N}_{\ge 0}\), a profit function \(p:V \rightarrow \mathbb{N}_{\ge 0}\), and an integer \(B\), find a set \(S\) of at most \(B\) edges not in \(E\) that maximises \(f(S)=\sum_{v\in V}w_v\cdot p(R(v,S))\), where \(p(R(v, S))\) is the sum of the profits of the nodes reachable from node \(v\) when the edges in \(S\) are added to \(G\). We first show that we can focus on Directed Acyclic Graphs (DAG) without loss of generality. We prove that the MCI problem on DAG is NP-Hard to approximate to within a factor greater than \(1-1/e\) even if we restrict to graphs with a single source or a single sink, and MCI remains NP-Complete if we further restrict to unitary weights. We devise a polynomial time algorithm based on dynamic programming to solve the MCI problem on trees with a single source. We propose a polynomial time greedy algorithm that guarantees \((1-1/e)\)-approximation ratio on DAGs with a single source or a single sink.
For the entire collection see [Zbl 1409.68018].Matroid-constrained vertex coverhttps://zbmath.org/1522.684062023-12-07T16:00:11.105023Z"Huang, Chien-Chung"https://zbmath.org/authors/?q=ai:huang.chien-chung"Sellier, François"https://zbmath.org/authors/?q=ai:sellier.francoisSummary: In this paper, we introduce the \textit{Matroid-Constrained Vertex Cover} problem: given a graph with weights on the edges and a matroid imposed on the vertices, our problem is to choose a subset of vertices that is independent in the matroid, with the objective of maximizing the total weight of covered edges. This problem is a generalization of the much studied \textsc{max} \(k\)-\textsc{vertex cover} problem, in which the matroid is the simple uniform matroid, and it is also a special case of the problem of maximizing a monotone submodular function under a matroid constraint.
In the first part of this work, we give a Fixed-Parameter Tractable Approximation Scheme (FPT-AS) when the given matroid is a partition matroid, a laminar matroid, or a transversal matroid. Precisely, if \(k\) is the rank of the matroid, we obtain \((1 - \varepsilon)\) approximation using \(( 1 / \varepsilon )^{O ( k )} n^{O ( 1 )}\) time for partition and laminar matroids and using \(( 1 / \varepsilon + k )^{O ( k )} n^{O ( 1 )}\) time for transversal matroids. This extends a result of Manurangsi for uniform matroids
[\textit{P. Manurangsi}, OASIcs -- OpenAccess Ser. Inform. 69, Paper No. 15, 21 p. (2018; \url{doi:10.4230/OASIcs.SOSA.2019.15})].
We also show that these ideas can be applied in the context of (single-pass) streaming algorithms. Besides, our FPT-AS introduces a new technique based on matroid union, which may be of independent interest in extremal combinatorics.
In the second part, we consider general matroids. We propose a simple local search algorithm that guarantees \(2 / 3 \approx 0.66\) approximation. For the more general problem where two matroids are imposed on the vertices and a feasible solution must be a common independent set, we show that a local search algorithm gives a \(2 / 3 \cdot ( 1 - 1 / ( p + 1 ) )\) approximation in \(n^{O ( p )}\) time, for any integer \(p\). We also provide some evidence to show that with the constraint of one or two matroids, the approximation ratio of 2/3 is likely the best possible, using the currently known techniques of local search.The upper bound on the Eulerian recurrent lengths of complete graphs obtained by an IP solverhttps://zbmath.org/1522.684082023-12-07T16:00:11.105023Z"Jimbo, Shuji"https://zbmath.org/authors/?q=ai:jimbo.shuji"Maruoka, Akira"https://zbmath.org/authors/?q=ai:maruoka.akiraSummary: If the degree of every vertex of a connected graph is even, then the graph has a circuit that contains all of edges, namely an Eulerian circuit. If the length of a shortest subcycle of an Eulerian circuit of a given graph is the largest, then the length is called the Eulerian recurrent length of the graph. For an odd integer \(n\) greater than or equal to \(3, e(n)\) denotes the Eulerian recurrent length of \(K_n\), the complete graph with \(n\) vertices. Values \(e(n)\) for all odd integers \(n\) with \(3\leqq n\leqq 13\) have been found by verification experiments using computers. If \(n\) is 7, 9, 11, or 13, then \(e(n) = n - 3\) holds, for example. On the other hand, it has been shown that \(n - 4\leqq e(n)\leqq n - 2\) holds for any odd integer \(n\) greater than or equal to 15 in previous researches. In this paper, it is proved that \(e(n)\leqq n - 3\) holds for every odd integer \(n\) greater than or equal to 15. In the core part of the proof of the main theorem, an IP (integer programming) solver is used as the amount of computation is too large to be solved by hand.
For the entire collection see [Zbl 1408.68014].On the tree augmentation problemhttps://zbmath.org/1522.684172023-12-07T16:00:11.105023Z"Nutov, Zeev"https://zbmath.org/authors/?q=ai:nutov.zeevSummary: In the Tree Augmentation problem we are given a tree \(T = (V, F)\) and a set \(E \subseteq V \times V\) of edges with positive integer costs \(\{c_e : e \in E\}\). The goal is to augment \(T\) by a minimum cost edge set \(J \subseteq E\) such that \(T \cup J\) is 2-edge-connected. We obtain the following results.
\begin{itemize}
\item[--] Recently, \textit{D. Adjiashvili} [SODA 2017, 2384--2399 (2017; Zbl 1410.68268)] introduced a novel LP for the problem and used it to break the 2-approximation barrier for instances when the maximum cost \(M\) of an edge in \(E\) is bounded by a constant; his algorithm computes a \(1.96418 + \epsilon\) approximate solution in time \(n^{{(M/\epsilon^2)}^{O(1)}}\). Using a simpler LP, we achieve ratio \(\frac{12}{7} + \epsilon\) in time \(2^{O(M/\epsilon^2)} \operatorname{poly}(n)\). This gives ratio better than 2 for logarithmic costs, and not only for constant costs.
\item[--] One of the oldest open questions for the problem is whether for unit costs (when \(M = 1)\) the standard LP-relaxation, so called Cut-LP, has integrality gap less than 2. We resolve this open question by proving that for unit costs the integrality gap of the Cut-LP is at most \(28/15 = 2 - 2/15\). In addition, we will prove that another natural LP-relaxation, that is much simpler than the ones in previous work, has integrality gap at most 7/4.
\end{itemize}Output space entropy search framework for multi-objective Bayesian optimizationhttps://zbmath.org/1522.684442023-12-07T16:00:11.105023Z"Belakaria, Syrine"https://zbmath.org/authors/?q=ai:belakaria.syrine"Deshwal, Aryan"https://zbmath.org/authors/?q=ai:deshwal.aryan"Doppa, Janardhan Rao"https://zbmath.org/authors/?q=ai:doppa.janardhan-raoSummary: We consider the problem of black-box multi-objective optimization (MOO) using expensive function evaluations (also referred to as experiments), where the goal is to approximate the true Pareto set of solutions by minimizing the total resource cost of experiments. For example, in hardware design optimization, we need to find the designs that trade-off performance, energy, and area overhead using expensive computational simulations. The key challenge is to select the sequence of experiments to uncover high-quality solutions using minimal resources. In this paper, we propose a general framework for solving MOO problems based on the principle of output space entropy (OSE) search: select the experiment that maximizes the information gained per unit resource cost about the true Pareto front. We appropriately instantiate the principle of OSE search to derive efficient algorithms for the following four MOO problem settings: 1) The most basic \textit{single-fidelity} setting, where experiments are expensive and accurate; 2) Handling \textit{black-box constraints} which cannot be evaluated without performing experiments; 3) The discrete \textit{multi-fidelity setting}, where experiments can vary in the amount of resources consumed and their evaluation accuracy; and 4) The \textit{continuous-fidelity setting}, where continuous function approximations result in a huge space of experiments. Experiments on diverse synthetic and real-world benchmarks show that our OSE search based algorithms improve over state-of-the-art methods in terms of both computational-efficiency and accuracy of MOO solutions.Jointly learning environments and control policies with projected stochastic gradient ascenthttps://zbmath.org/1522.684462023-12-07T16:00:11.105023Z"Bolland, Adrien"https://zbmath.org/authors/?q=ai:bolland.adrien"Boukas, Ioannis"https://zbmath.org/authors/?q=ai:boukas.ioannis"Berger, Mathias"https://zbmath.org/authors/?q=ai:berger.mathias"Ernst, Damien"https://zbmath.org/authors/?q=ai:ernst.damienSummary: We consider the joint design and control of discrete-time stochastic dynamical systems over a finite time horizon. We formulate the problem as a multi-step optimization problem under uncertainty seeking to identify a system design and a control policy that jointly maximize the expected sum of rewards collected over the time horizon considered. The transition function, the reward function and the policy are all parametrized, assumed known and differentiable with respect to their parameters. We then introduce a deep reinforcement learning algorithm combining policy gradient methods with model-based optimization techniques to solve this problem. In essence, our algorithm iteratively approximates the gradient of the expected return via Monte-Carlo sampling and automatic differentiation and takes projected gradient ascent steps in the space of environment and policy parameters. This algorithm is referred to as Direct Environment and Policy Search (DEPS). We assess the performance of our algorithm in three environments concerned with the design and control of a mass-spring-damper system, a small-scale off-grid power system and a drone, respectively. In addition, our algorithm is benchmarked against a state-of-the-art deep reinforcement learning algorithm used to tackle joint design and control problems. We show that DEPS performs at least as well or better in all three environments, consistently yielding solutions with higher returns in fewer iterations. Finally, solutions produced by our algorithm are also compared with solutions produced by an algorithm that does not jointly optimize environment and policy parameters, highlighting the fact that higher returns can be achieved when joint optimization is performed.Task-aware verifiable RNN-based policies for partially observable Markov decision processeshttps://zbmath.org/1522.684482023-12-07T16:00:11.105023Z"Carr, Steven"https://zbmath.org/authors/?q=ai:carr.steven"Jansen, Nils"https://zbmath.org/authors/?q=ai:jansen.nils"Topcu, Ufuk"https://zbmath.org/authors/?q=ai:topcu.ufukSummary: Partially observable Markov decision processes (POMDPs) are models for sequential decision-making under uncertainty and incomplete information. Machine learning methods typically train recurrent neural networks (RNN) as effective representations of POMDP policies that can efficiently process sequential data. However, it is hard to verify whether the POMDP driven by such RNN-based policies satisfies safety constraints, for instance, given by temporal logic specifications. We propose a novel method that combines techniques from machine learning with the field of formal methods: training an RNN-based policy and then automatically extracting a so-called finite-state controller (FSC) from the RNN. Such FSCs offer a convenient way to verify temporal logic constraints. Implemented on a POMDP, they induce a Markov chain, and probabilistic verification methods can efficiently check whether this induced Markov chain satisfies a temporal logic specification. Using such methods, if the Markov chain does not satisfy the specification, a byproduct of verification is diagnostic information about the states in the POMDP that are critical for the specification. The method exploits this diagnostic information to either adjust the complexity of the extracted FSC or improve the policy by performing focused retraining of the RNN. The method synthesizes policies that satisfy temporal logic specifications for POMDPs with up to millions of states, which are three orders of magnitude larger than comparable approaches.Efficient implicit Lagrangian twin parametric insensitive support vector regression via unconstrained minimization problemshttps://zbmath.org/1522.684562023-12-07T16:00:11.105023Z"Gupta, Deepak"https://zbmath.org/authors/?q=ai:gupta.deepak-kumar"Richhariya, Bharat"https://zbmath.org/authors/?q=ai:richhariya.bharatSummary: In this paper, an efficient implicit Lagrangian twin parametric insensitive support vector regression is proposed which leads to a pair of unconstrained minimization problems, motivated by the works on twin parametric insensitive support vector regression
[\textit{X. Peng}, Neurocomputing 79, 26--38 (2012; \url{doi:10.1016/j.neucom.2011.09.021})], and Lagrangian twin support vector regression [\textit{S. Balasundaram} and \textit{M. Tanveer}, Neural Comput. Appl. 22, No. 1, 257--267 (2013; \url{doi:10.1007/s00521-012-0971-9})]. Since its objective function is strongly convex, piece-wise quadratic and differentiable, it can be solved by gradient-based iterative methods. Notice that its objective function having non-smooth `plus' function, so one can consider either generalized Hessian, or smooth approximation function to replace the `plus' function and further apply the simple Newton-Armijo step size algorithm. These algorithms can be easily implemented in MATLAB and do not require any optimization toolbox. The advantage of this method is that proposed algorithms take less training time and can deal with data having heteroscedastic noise structure. To demonstrate the effectiveness of the proposed method, computational results are obtained on synthetic and real-world datasets which clearly show comparable generalization performance and improved learning speed in accordance with support vector regression, twin support vector regression, and twin parametric insensitive support vector regression.Classification with costly features as a sequential decision-making problemhttps://zbmath.org/1522.684622023-12-07T16:00:11.105023Z"Janisch, Jaromír"https://zbmath.org/authors/?q=ai:janisch.jaromir"Pevný, Tomáš"https://zbmath.org/authors/?q=ai:pevny.tomas"Lisý, Viliam"https://zbmath.org/authors/?q=ai:lisy.viliamSummary: This work focuses on a specific classification problem, where the information about a sample is not readily available, but has to be acquired for a cost, and there is a per-sample budget. Inspired by real-world use-cases, we analyze \textit{average} and \textit{hard} variations of a \textit{directly specified} budget. We postulate the problem in its explicit formulation and then convert it into an equivalent MDP, that can be solved with deep reinforcement learning. Also, we evaluate a real-world inspired setting with sparse training datasets with missing features. The presented method performs robustly well in all settings across several distinct datasets, outperforming other prior-art algorithms. The method is flexible, as showcased with all mentioned modifications and can be improved with any domain independent advancement in RL.Breaking the sample complexity barrier to regret-optimal model-free reinforcement learninghttps://zbmath.org/1522.684732023-12-07T16:00:11.105023Z"Li, Gen"https://zbmath.org/authors/?q=ai:li.gen.1"Shi, Laixi"https://zbmath.org/authors/?q=ai:shi.laixi"Chen, Yuxin"https://zbmath.org/authors/?q=ai:chen.yuxin"Chi, Yuejie"https://zbmath.org/authors/?q=ai:chi.yuejieSummary: Achieving sample efficiency in online episodic reinforcement learning (RL) requires optimally balancing exploration and exploitation. When it comes to a finite-horizon episodic Markov decision process with \(S\) states, \(A\) actions and horizon length \(H\), substantial progress has been achieved toward characterizing the minimax-optimal regret, which scales on the order of \(\sqrt{H^2SAT}\) (modulo log factors) with \(T\) the total number of samples. While several competing solution paradigms have been proposed to minimize regret, they are either memory-inefficient, or fall short of optimality unless the sample size exceeds an enormous threshold (e.g. \(S^6A^4\operatorname{poly}(H)\) for existing model-free methods).
To overcome such a large sample size barrier to efficient RL, we design a novel model-free algorithm, with space complexity \(O(SAH)\), that achieves near-optimal regret as soon as the sample size exceeds the order of \(SA\operatorname{poly}(H)\). In terms of this sample size requirement (also referred to the initial burn-in cost), our method improves -- by at least a factor of \(S^5A^3\) -- upon any prior memory-efficient algorithm that is asymptotically regret-optimal. Leveraging the recently introduced variance reduction strategy (also called \textit{reference-advantage decomposition}), the proposed algorithm employs an \textit{early-settled} reference update rule, with the aid of two Q-learning sequences with upper and lower confidence bounds. The design principle of our early-settled variance reduction method might be of independent interest to other RL settings that involve intricate exploration-exploitation trade-offs.Solving the traveling salesperson problem with precedence constraints by deep reinforcement learninghttps://zbmath.org/1522.684992023-12-07T16:00:11.105023Z"Löwens, Christian"https://zbmath.org/authors/?q=ai:lowens.christian"Ashraf, Inaam"https://zbmath.org/authors/?q=ai:ashraf.inaam"Gembus, Alexander"https://zbmath.org/authors/?q=ai:gembus.alexander"Cuizon, Genesis"https://zbmath.org/authors/?q=ai:cuizon.genesis"Falkner, Jonas K."https://zbmath.org/authors/?q=ai:falkner.jonas-k"Schmidt-Thieme, Lars"https://zbmath.org/authors/?q=ai:schmidt-thieme.larsSummary: This work presents solutions to the Traveling Salesperson Problem with precedence constraints (TSPPC) using Deep Reinforcement Learning (DRL) by adapting recent approaches that work well for regular TSPs. Common to these approaches is the use of graph models based on multi-head attention layers. One idea for solving the pickup and delivery problem (PDP) is using heterogeneous attentions to embed the different possible roles each node can take. In this work, we generalize this concept of heterogeneous attentions to the TSPPC. Furthermore, we adapt recent ideas to sparsify attentions for better scalability. Overall, we contribute to the research community through the application and evaluation of recent DRL methods in solving the TSPPC. Our code is available at \url{https://github.com/christianll9/tsppc-drl}.
For the entire collection see [Zbl 1511.68009].Steady-state planning in expected reward multichain MDPshttps://zbmath.org/1522.685122023-12-07T16:00:11.105023Z"Atia, George K."https://zbmath.org/authors/?q=ai:atia.george-k"Beckus, Andre"https://zbmath.org/authors/?q=ai:beckus.andre"Alkhouri, Ismail"https://zbmath.org/authors/?q=ai:alkhouri.ismail"Velasquez, Alvaro"https://zbmath.org/authors/?q=ai:velasquez.alvaroSummary: The planning domain has experienced increased interest in the formal synthesis of decision-making policies. This formal synthesis typically entails finding a policy which satisfies formal specifications in the form of some well-defined logic. While many such logics have been proposed with varying degrees of expressiveness and complexity in their capacity to capture desirable agent behavior, their value is limited when deriving decision-making policies which satisfy certain types of asymptotic behavior in general system models. In particular, we are interested in specifying constraints on the steady-state behavior of an agent, which captures the proportion of time an agent spends in each state as it interacts for an indefinite period of time with its environment. This is sometimes called the average or expected behavior of the agent and the associated planning problem is faced with significant challenges unless strong restrictions are imposed on the underlying model in terms of the connectivity of its graph structure. In this paper, we explore this steady-state planning problem that consists of deriving a decision-making policy for an agent such that constraints on its steady-state behavior are satisfied. A linear programming solution for the general case of multichain Markov Decision Processes (MDPs) is proposed and we prove that optimal solutions to the proposed programs yield stationary policies with rigorous guarantees of behavior.A linear constrained optimization Benchmark for probabilistic search algorithms: the rotated Klee-Minty problemhttps://zbmath.org/1522.685162023-12-07T16:00:11.105023Z"Hellwig, Michael"https://zbmath.org/authors/?q=ai:hellwig.michael"Beyer, Hans-Georg"https://zbmath.org/authors/?q=ai:beyer.hans-georgSummary: The development, assessment, and comparison of randomized search algorithms heavily rely on benchmarking. Regarding the domain of constrained optimization, the small number of currently available benchmark environments bears no relation to the vast number of distinct problem features. The present paper advances a proposal of a scalable linear constrained optimization problem that is suitable for benchmarking Evolutionary Algorithms. By comparing two recent Evolutionary Algorithm variants, the linear benchmarking environment is demonstrated.
For the entire collection see [Zbl 1407.68029].Landscape-aware constraint handling applied to differential evolutionhttps://zbmath.org/1522.685172023-12-07T16:00:11.105023Z"Malan, Katherine M."https://zbmath.org/authors/?q=ai:malan.katherine-mSummary: In real-world contexts optimisation problems frequently have constraints. Evolutionary algorithms do not naturally handle constrained spaces, so require constraint handling techniques to modify the search process. Based on the thesis that different constraint handling approaches are suited to different problem types, this study shows that the features of the problem can provide guidance in choosing appropriate constraint handling techniques for differential evolution. High level algorithm selection rules are derived through data mining based on a training set of problems on which landscape analysis is performed through sampling. On a set of different test problems, these rules are used to switch between constraint handling techniques during differential evolution search using on-line analysis of landscape features. The proposed landscape-aware switching approach is shown to out-perform the constituent constraint-handling approaches, illustrating that there is value in monitoring the landscape during search and switching to appropriate techniques depending on the problem characteristics. Results are also provided that show that the approach is fairly insensitive to parameter changes.
For the entire collection see [Zbl 1407.68029].An interdisciplinary experimental evaluation on the disjunctive temporal problemhttps://zbmath.org/1522.685192023-12-07T16:00:11.105023Z"Zavatteri, Matteo"https://zbmath.org/authors/?q=ai:zavatteri.matteo"Raffaele, Alice"https://zbmath.org/authors/?q=ai:raffaele.alice"Ostuni, Dario"https://zbmath.org/authors/?q=ai:ostuni.dario"Rizzi, Romeo"https://zbmath.org/authors/?q=ai:rizzi.romeoSummary: We report on an extensive experimental evaluation on the Disjunctive Temporal Problem, where we adapted state-of-the-art Satisfiability Modulo Theories (SMT) encodings into the frameworks of Mixed Integer Linear Programming (MILP), (Circuit) Satisfiability (SAT), and Constraint Programming (CP). We considered 6 SMT solvers, 4 MILP solvers, 3 SAT solvers, and 3 CP solvers, broadly-recognized for their technologies. We compared all of them on several sets of benchmarks. As well as considering 2 random sets in the literature, we generated 3 new industrial and 3 new computationally-hard sets of benchmarks, which we make publicly available online. In particular, we analyzed 9885 instances, each processed on average about 33 times. Overall, SMT is confirmed to be the current best technology, but also MILP can perform very well, for instance on some random instances, on which it can be up to 2x faster than SMT. On a single machine, this experimental evaluation would have taken 598.97 days.Fast semi-supervised evidential clusteringhttps://zbmath.org/1522.685472023-12-07T16:00:11.105023Z"Antoine, Violaine"https://zbmath.org/authors/?q=ai:antoine.violaine"Guerrero, Jose A."https://zbmath.org/authors/?q=ai:guerrero.jose-atilio|guerrero.jose-alfredo"Xie, Jiarui"https://zbmath.org/authors/?q=ai:xie.jiaruiSummary: Semi-supervised clustering is a constrained clustering technique that organizes a collection of unlabeled data into homogeneous subgroups with the help of domain knowledge expressed as constraints. These methods are, most of the time, variants of the popular \textit{\(k\)-means} clustering algorithm. As such, they are based on a criterion to minimize. Amongst existing semi-supervised clusterings, Semi-supervised Evidential Clustering (SECM) deals with the problem of uncertain/imprecise labels and creates a credal partition. In this work, a new heuristic algorithm, called SECM-h, is presented. The proposed algorithm relaxes the constraints of SECM in such a way that the optimization problem is solved using the Lagrangian method. Experimental results show that the proposed algorithm largely improves execution time while accuracy is maintained.Improving and benchmarking of algorithms for \(\Gamma \)-maximin, \( \Gamma \)-maximax and interval dominancehttps://zbmath.org/1522.685792023-12-07T16:00:11.105023Z"Nakharutai, Nawapon"https://zbmath.org/authors/?q=ai:nakharutai.nawapon"Troffaes, Matthias C. M."https://zbmath.org/authors/?q=ai:troffaes.matthias-c-m"Caiado, Camila C. S."https://zbmath.org/authors/?q=ai:caiado.camila-c-sSummary: \( \Gamma \)-maximin, \( \Gamma \)-maximax and interval dominance are familiar decision criteria for making decisions under severe uncertainty, when probability distributions can only be partially identified. One can apply these three criteria by solving sequences of linear programs. In this study, we present new algorithms for these criteria and compare their performance to existing standard algorithms. Specifically, we use efficient ways, based on previous work, to find common initial feasible points for these algorithms. Exploiting these initial feasible points, we develop early stopping criteria to determine whether gambles are either \(\Gamma \)-maximin, \( \Gamma \)-maximax or interval dominant. We observe that the primal-dual interior point method benefits considerably from these improvements. In our simulation, we find that our proposed algorithms outperform the standard algorithms when the size of the domain of lower previsions is less or equal to the sizes of decisions and outcomes. However, our proposed algorithms do not outperform the standard algorithms in the case that the size of the domain of lower previsions is much larger than the sizes of decisions and outcomes.On the problem of possibilistic-probabilistic optimization with constraints on possibility/probabilityhttps://zbmath.org/1522.685882023-12-07T16:00:11.105023Z"Yazenin, Alexander"https://zbmath.org/authors/?q=ai:yazenin.aleksander-vasilevich"Soldatenko, Ilia"https://zbmath.org/authors/?q=ai:soldatenko.ilia-s|soldatenko.iliaSummary: The problem of possibilistic-probabilistic linear programming with constraints on possibility and probability is investigated. For the case of normally distributed random parameters of the model and under the most general assumptions concerning the properties of probability distributions, its equivalent determinate analogue is constructed which is the quadratic programming model.
For the entire collection see [Zbl 1409.68016].Order preserving barrier coverage with weighted sensors on a linehttps://zbmath.org/1522.686302023-12-07T16:00:11.105023Z"Benkoczi, Robert"https://zbmath.org/authors/?q=ai:benkoczi.robert-r"Gaur, Daya"https://zbmath.org/authors/?q=ai:gaur.daya"Zhang, Xiao"https://zbmath.org/authors/?q=ai:zhang.xiao.2Summary: We consider a barrier coverage problem with heterogeneous mobile sensors where the sensors are located on a line and the goal is to move the sensors so that a given line segment, called the barrier, is covered by the sensors. We focus on an important generalization of the classical model where the cost of moving a sensor equals the weighted travel distance and the sensor weights are arbitrary. For the objective of minimizing the maximum cost of moving a sensor, it was recently shown that the problem is NP-hard when the sensing ranges are arbitrary. In contrast, Chen et al. give an \(O(n^2 \log n)\) algorithm for the problem with uniform weights and arbitrary sensing ranges. Xiao shows that restricting the problem to uniform sensing ranges but allowing arbitrary sensor weights can also be solved exactly in time \(O(n^2 \log n \log \log n)\), raising the question whether other restrictions can be solved in time polynomial in the size of the instance. In this paper, we show that a natural restriction in which sensors must preserve their relative ordering (the sensors move on rails, for example) but the sensors have arbitrary sensing ranges and weights can be solved in time \((n^2 \log^3 n)\). Due to the combinatorially rich set of configurations of the optimal solution for our problem, our algorithm uses the general parametric search method of Megiddo which parameterizes a feasibility test algorithm. Interestingly, it is not easy to design an efficient feasibility test algorithm for the order preserving problem. We overcome the difficulties using the concept of critical budget values and employing standard computational geometry techniques.
For the entire collection see [Zbl 1400.68037].Fundamental domains for symmetric optimization: construction and searchhttps://zbmath.org/1522.686412023-12-07T16:00:11.105023Z"Danielson, Claus"https://zbmath.org/authors/?q=ai:danielson.clausSummary: Symmetries of a set are linear transformations that map the set to itself. A fundamental domain of a symmetric set is a subset that contains at least one representative from each of the symmetric equivalence classes (orbits) in the set. This paper contributes a novel polynomial algorithm for constructing minimal polytopic fundamental domains of polytopic sets. Our algorithm is applicable for generic linear symmetries of the set and has linear complexity in the number of facets and dimension of the symmetric polytope, e.g., the feasible region of an optimization problem. In addition, this paper contributes a novel polynomial algorithm for mapping an element of the polytope to its representative in the fundamental domain. The algorithms are demonstrated in four examples -- two illustrative and two practical. In the first practical example, we show that a minimal fundamental domain of the hypercube under the symmetric group is the set of points with sorted elements. In the second practical example, we show how the construction algorithm can be applied to the max-cut problem.Digital geometry, mathematical morphology, and discrete optimization: a surveyhttps://zbmath.org/1522.686562023-12-07T16:00:11.105023Z"Kiselman, Christer Oscar"https://zbmath.org/authors/?q=ai:kiselman.christer-oscarSummary: We study difficulties that appear when well-established definitions and results in Euclidean geometry, especially in the theory of convex sets and functions in vector spaces, are translated into a discrete setting. Solutions to these problems are sketched.
For the entire collection see [Zbl 1511.68008].Structured discrete shape approximation: theoretical complexity and practical algorithmhttps://zbmath.org/1522.686812023-12-07T16:00:11.105023Z"Tillmann, Andreas M."https://zbmath.org/authors/?q=ai:tillmann.andreas-m"Kobbelt, Leif"https://zbmath.org/authors/?q=ai:kobbelt.leifSummary: We consider the problem of approximating a two-dimensional shape contour (or curve segment) using discrete assembly systems, which allow to build geometric structures based on limited sets of node and edge types subject to edge length and orientation restrictions. We show that already deciding feasibility of such approximation problems is \textsf{NP}-hard, and remains intractable even for very simple setups. We then devise an algorithmic framework that combines shape sampling with exact cardinality minimization to obtain good approximations using few components. As a particular application and showcase example, we discuss approximating shape contours using the classical Zometool construction kit and provide promising computational results, demonstrating that our algorithm is capable of obtaining good shape representations within reasonable time, in spite of the problem's general intractability. We conclude the paper with an outlook on possible extensions of the developed methodology, in particular regarding 3D shape approximation tasks.Handling polynomial and transcendental functions in SMT via unconstrained optimisation and topological degree testhttps://zbmath.org/1522.687162023-12-07T16:00:11.105023Z"Cimatti, Alessandro"https://zbmath.org/authors/?q=ai:cimatti.alessandro"Griggio, Alberto"https://zbmath.org/authors/?q=ai:griggio.alberto"Lipparini, Enrico"https://zbmath.org/authors/?q=ai:lipparini.enrico"Sebastiani, Roberto"https://zbmath.org/authors/?q=ai:sebastiani.robertoSummary: We present a method for determining the satisfiability of quantifier-free first-order formulas modulo the theory of non-linear arithmetic over the reals augmented with transcendental functions. Our procedure is based on the fruitful combination of two main ingredients: unconstrained optimisation, to generate a set of candidate solutions, and a result from topology called the topological degree test to check whether a given bounded region contains at least a solution. We have implemented the procedure in a prototype tool called \textsc{ugotNL}, and integrated it within the \textsc{MathSAT} SMT solver. Our experimental evaluation over a wide range of benchmarks shows that it vastly improves the performance of the solver for satisfiable non-linear arithmetic formulas, significantly outperforming other available tools for problems with transcendental functions.
For the entire collection see [Zbl 1511.68011].Partial resampling to approximate covering integer programshttps://zbmath.org/1522.687542023-12-07T16:00:11.105023Z"Chen, Antares"https://zbmath.org/authors/?q=ai:chen.antares"Harris, David G."https://zbmath.org/authors/?q=ai:harris.david-g"Srinivasan, Aravind"https://zbmath.org/authors/?q=ai:srinivasan.aravindSummary: We consider column-sparse covering integer programs, a generalization of set cover. We develop a new rounding scheme based on the partial resampling variant of the Lovász Local Lemma developed by Harris and Srinivasan. This achieves an approximation ratio of \[1 + \frac{\ln (\Delta_1 + 1)}{a_{\min}} + O\left( \log(1+\sqrt{\frac{\log (\Delta_1 +1)}{a_{\min}}})\right),\] where \(a_{\min}\) is the minimum covering constraint and \(\Delta_1\) is the maximum \(\ell_1\)-norm of any column of the covering matrix \(A\) (whose entries are scaled to lie in \([0, 1]\)). With additional constraints on the variable sizes, we get an approximation ratio of \(\ln \Delta_0 + O(\log \log \Delta_0)\) (where \(\Delta_0\) is the maximum number of nonzero entries in any column of \(A\)). These results improve asymptotically over results of Srinivasan and results of Kolliopoulos and Young. We show nearly-matching lower bounds. We also show that the rounding process leads to negative correlation among the variables.
{\copyright} 2020 Wiley Periodicals LLCA new and improved algorithm for online bin packinghttps://zbmath.org/1522.687582023-12-07T16:00:11.105023Z"Balogh, János"https://zbmath.org/authors/?q=ai:balogh.janos"Békési, József"https://zbmath.org/authors/?q=ai:bekesi.jozsef"Dósa, György"https://zbmath.org/authors/?q=ai:dosa.gyorgy"Epstein, Leah"https://zbmath.org/authors/?q=ai:epstein.leah"Levin, Asaf"https://zbmath.org/authors/?q=ai:levin.asafSummary: We revisit the classic online bin packing problem studied in the half-century. In this problem, items of positive sizes no larger than 1 are presented one by one to be packed into subsets called bins of total sizes no larger than 1, such that every item is assigned to a bin before the next item is presented. We use online partitioning of items into classes based on sizes, as in previous work, but we also apply a new method where items of one class can be packed into more than two types of bins, where a bin type is defined according to the number of such items grouped together. Additionally, we allow the smallest class of items to be packed in multiple kinds of bins, and not only into their own bins. We combine this with the approach of packing of sufficiently big items according to their exact sizes. Finally, we simplify the analysis of such algorithms, allowing the analysis to be based on the most standard weight functions. This simplified analysis allows us to study the algorithm which we defined based on all these ideas. This leads us to the design and analysis of the first algorithm of asymptotic competitive ratio strictly below 1.58, specifically, we break this barrier by providing an algorithm AH (Advanced Harmonic) whose asymptotic competitive ratio does not exceed 1.57829.\par Our main contribution is the introduction of the simple analysis based on weight function to analyze the state of the art online algorithms for the classic online bin packing problem. The previously used analytic tool named weight system was too complicated for the community in this area to adjust it for other problems and other algorithmic tools that are needed in order to improve the current best algorithms. We show that the weight system based analysis is not needed for the analysis of the current algorithms for the classic online bin packing problem. The importance of a simple analysis is demonstrated by analyzing several new features together with all existing techniques, and by proving a better competitive ratio than the previously best one.
For the entire collection see [Zbl 1393.68010].Improved streaming algorithms for maximizing monotone submodular functions under a knapsack constrainthttps://zbmath.org/1522.687612023-12-07T16:00:11.105023Z"Huang, Chien-Chung"https://zbmath.org/authors/?q=ai:huang.chien-chung"Kakimura, Naonori"https://zbmath.org/authors/?q=ai:kakimura.naonoriSummary: In this paper, we consider the problem of maximizing a monotone submodular function subject to a knapsack constraint in a streaming setting. In such a setting, elements arrive sequentially and at any point in time, and the algorithm can store only a small fraction of the elements that have arrived so far. For the special case that all elements have unit sizes (i.e., the cardinality-constraint case), one can find a \((0.5 - \varepsilon)\)-approximate solution in \(O(K \varepsilon^{-1})\) space, where \(K\) is the knapsack capacity
[\textit{A. Badanidiyuru} et al., in: Proceedings of the 20th ACM SIGKDD international conference on knowledge discovery and data mining, KDD'14. New York, NY: Association for Computing Machinery (ACM). 671--680 (2014; \url{doi:10.1145/2623330.2623637})].
The approximation ratio is recently shown to be optimal
[\textit{M. Feldman} et al., in: Proceedings of the 52nd annual ACM SIGACT symposium on theory of computing, STOC'20. New York, NY: Association for Computing Machinery (ACM). 1363--1374 (2020; Zbl 07298334)].
In this work, we propose a \((0.4 - \varepsilon)\)-approximation algorithm for the knapsack-constrained problem, using space that is a polynomial of \(K\) and \(\varepsilon\). This improves on the previous best ratio of \(0.363 - \varepsilon\) with space of the same order. Our algorithm is based on a careful combination of various ideas to transform multiple-pass streaming algorithms into a single-pass one.Linear pseudo-polynomial factor algorithm for automaton constrained tree knapsack problemhttps://zbmath.org/1522.687702023-12-07T16:00:11.105023Z"Kumabe, Soh"https://zbmath.org/authors/?q=ai:kumabe.soh"Maehara, Takanori"https://zbmath.org/authors/?q=ai:maehara.takanori"Sin'ya, Ryoma"https://zbmath.org/authors/?q=ai:sinya.ryomaSummary: The \textit{automaton constrained tree knapsack problem} is a variant of the knapsack problem in which the items are associated with the vertices of the tree, and we can select a subset of items that is accepted by a tree automaton. If the capacities or the profits of items are integers, it can be solved in pseudo-polynomial time by the dynamic programming algorithm. However, this algorithm has a quadratic pseudo-polynomial factor in its complexity because of the max-plus convolution. In this study, we propose a new dynamic programming technique, called \textit{heavy-light recursive dynamic programming}, to obtain algorithms having linear pseudo-polynomial factors in the complexity. Such algorithms can be used for solving the problems with polynomially small capacities/profits efficiently, and used for deriving efficient fully polynomial-time approximation schemes. We also consider the \(k\)-subtree version problem that finds \(k\) disjoint subtrees and a solution in each subtree that maximizes total profit under a budget constraint. We show that this problem can be solved in almost the same complexity as the original problem.
For the entire collection see [Zbl 1408.68014].A fast algorithm for unbounded monotone integer linear systems with two variables per inequality via graph decompositionhttps://zbmath.org/1522.687722023-12-07T16:00:11.105023Z"Tamori, Takuya"https://zbmath.org/authors/?q=ai:tamori.takuya"Kimura, Kei"https://zbmath.org/authors/?q=ai:kimura.keiSummary: In this paper, we consider the feasibility problem of integer linear systems where each inequality has at most two variables. Although the problem is known to be weakly NP-complete by Lagarias, it has many applications and, importantly, a large subclass of it admits (pseudo-)polynomial algorithms. Indeed, the problem is shown pseudo-polynomially solvable if every variable has upper and lower bounds by Hochbaum, Megiddo, Naor, and Tamir. However, determining the complexity of the general case, pseudo-polynomially solvable or strongly NP-complete, is a longstanding open problem. In this paper, we reveal a new efficiently solvable subclass of the problem. Namely, for the \textit{monotone} case, i.e., when two coefficients of the two variables in each inequality are opposite signs, we associate a directed graph to any instance, and present an algorithm that runs in \(O(n \cdot s \cdot 2^{O(\ell \log \ell )} + n + m)\) time, where \(s\) is the length of the input and \(\ell\) is the maximum number of the vertices in any strongly connected component of the graph. If \(\ell\) is a constant, the algorithm runs in polynomial time. From the result, it can be observed that the hardness of the feasibility problem lies on large strongly connected components of the graph.
For the entire collection see [Zbl 1408.68014].The design of (almost) disjunct matrices by evolutionary algorithmshttps://zbmath.org/1522.687742023-12-07T16:00:11.105023Z"Knezevic, Karlo"https://zbmath.org/authors/?q=ai:knezevic.karlo"Picek, Stjepan"https://zbmath.org/authors/?q=ai:picek.stjepan"Mariot, Luca"https://zbmath.org/authors/?q=ai:mariot.luca"Jakobovic, Domagoj"https://zbmath.org/authors/?q=ai:jakobovic.domagoj"Leporati, Alberto"https://zbmath.org/authors/?q=ai:leporati.albertoSummary: Disjunct Matrices (DM) are a particular kind of binary matrices which have been especially applied to solve the Non-Adaptive Group Testing (NAGT) problem, where the task is to detect any configuration of \(t\) defectives out of a population of \(N\) items. Traditionally, the methods used to construct DM leverage on error-correcting codes and other related algebraic techniques. Here, we investigate the use of Evolutionary Algorithms to design DM and two of their generalizations, namely Resolvable Matrices (RM) and Almost Disjunct Matrices (ADM). After discussing the basic encoding used to represent the candidate solutions of our optimization problems, we define three fitness functions, each measuring the deviation of a generic binary matrix from being respectively a DM, an RM or an ADM. Next, we employ Estimation of Distribution Algorithms (EDA), Genetic Algorithms (GA), and Genetic Programming (GP) to optimize these fitness functions. The results show that GP achieves the best performances among the three heuristics, converging to an optimal solution on a wider range of problem instances. Although these results do not match those obtained by other state-of-the-art methods in the literature, we argue that our heuristic approach can generate solutions that are not expressible by currently known algebraic techniques, and sketch some possible ideas to further improve its performance.
For the entire collection see [Zbl 1407.68029].Classical and quantum random-walk centrality measures in multilayer networkshttps://zbmath.org/1522.810992023-12-07T16:00:11.105023Z"Böttcher, Lucas"https://zbmath.org/authors/?q=ai:bottcher.lucas"Porter, Mason A."https://zbmath.org/authors/?q=ai:porter.mason-aSummary: Multilayer network analysis is a useful approach for studying networks of entities that interact with each other via multiple relationships. Classifying the importance of nodes and node-layer tuples is an important aspect of the study of multilayer networks. To do this, it is common to calculate various centrality measures, which allow one to rank nodes and node-layers according to a variety of structural features. In this paper, we formulate occupation, PageRank, betweenness, and closeness centralities in terms of node-occupation properties of different types of continuous-time classical and quantum random walks on multilayer networks. We apply our framework to a variety of synthetic and real-world multilayer networks, and we identify notable differences between classical and quantum centrality measures. Our computations give insights into the correlations between certain centralities that are based on random walks and associated centralities that are based on geodesic paths.Algorithms for convex optimizationhttps://zbmath.org/1522.900012023-12-07T16:00:11.105023Z"Vishnoi, Nisheeth K."https://zbmath.org/authors/?q=ai:vishnoi.nisheeth-kPublisher's description: In the last few years, Algorithms for Convex Optimization have revolutionized algorithm design, both for discrete and continuous optimization problems. For problems like maximum flow, maximum matching, and submodular function minimization, the fastest algorithms involve essential methods such as gradient descent, mirror descent, interior point methods, and ellipsoid methods. The goal of this self-contained book is to enable researchers and professionals in computer science, data science, and machine learning to gain an in-depth understanding of these algorithms. The text emphasizes how to derive key algorithms for convex optimization from first principles and how to establish precise running time bounds. This modern text explains the success of these algorithms in problems of discrete optimization, as well as how these methods have significantly pushed the state of the art of convex optimization itself.Monitoring a fleet of autonomous vehicles through A* like algorithms and reinforcement learninghttps://zbmath.org/1522.900022023-12-07T16:00:11.105023Z"Baiou, Mourad"https://zbmath.org/authors/?q=ai:baiou.mourad"Mombelli, Aurélien"https://zbmath.org/authors/?q=ai:mombelli.aurelien"Quilliot, Alain"https://zbmath.org/authors/?q=ai:quilliot.alainSummary: We deal here with a fleet of autonomous vehicles which is required to perform internal logistics tasks inside some protected area. This fleet is supposed to be ruled by a hierarchical supervision architecture, which, at the top level distributes and schedules Pick up and Delivery tasks, and, at the lowest level, ensures safety at the crossroads and controls the trajectories. We focus here on the top level, while introducing a time dependent estimation of the risk induced by the traversal of any arc at a given time. We set a model, state some structural results, and design, in order to route and schedule the vehicles according to a well-fitted compromise between speed and risk, a bi-level algorithm and a A* algorithm which both relies on a reinforcement learning scheme.
For the entire collection see [Zbl 1498.90005].More-for-less solutions in fuzzy transportation problemshttps://zbmath.org/1522.900032023-12-07T16:00:11.105023Z"Bhatia, Tanveen Kaur"https://zbmath.org/authors/?q=ai:bhatia.tanveen-kaur"Kumar, Amit"https://zbmath.org/authors/?q=ai:kumar.amit.1"Appadoo, Srimantoorao S."https://zbmath.org/authors/?q=ai:appadoo.srimantoorao-sPublisher's description: This book describes a set of methods for finding more-for-less solutions of various kind of fuzzy transportation problems. Inspired by more-for-less approaches to the basic transportation problem initiated by Abraham Charnes and his collaborators during 1960s and 1970s, this book describes new methods developed by the authors to solve different types of problems, including symmetric balanced fuzzy transportation problems, symmetric intuitionistic fuzzy transportation problems with mixed constraints, and symmetric intuitionistic fuzzy linear fractional transportation problems with mixed constraints. It offers extensive details on their applications to some representative problems, and discusses some future research directions.A multi-objective robust possibilistic programming approach to sustainable public transportation network designhttps://zbmath.org/1522.900042023-12-07T16:00:11.105023Z"Günay, Elif Elçin"https://zbmath.org/authors/?q=ai:gunay.elif-elcin"Okudan Kremer, Gül E."https://zbmath.org/authors/?q=ai:okudan-kremer.gul-e"Zarindast, Atousa"https://zbmath.org/authors/?q=ai:zarindast.atousaSummary: As a critical component of sustainable development, a transportation system should be designed such that it has a positive impact on the economic, environmental, and social sustainability of the served region. In response, this study introduces the concept of passenger dissatisfaction with additional walking and waiting as an indicator of social sustainability and uses the concept while optimizing the transit network for economic, environmental, and social perfectives. Due to a lack of knowledge about the actual value of different passenger dissatisfaction levels and uncertainty in demand, a multi-objective robust possibilistic programming approach \((RPP^\ast)\) is proposed and solved by using an interactive fuzzy programming approach. Different from other robust possibilistic approaches, \(RPP^\ast\) optimizes not only the mean of the objective function and chance constraint violations but also the risk value inherited by uncertain parameters through considering the absolute deviation of the objective function. Both the advantage of \(RPP^\ast\) versus the deterministic model and its superiority against several robust possibilistic approaches are demonstrated in the numerical studies. Furthermore, the outcomes of the numerical study demonstrate that the transportation network should be designed in a decentralized way as the risk coefficients, i.e., risk-taking attitude, increase.The paired mail carrier problemhttps://zbmath.org/1522.900052023-12-07T16:00:11.105023Z"Luo, Yuchen"https://zbmath.org/authors/?q=ai:luo.yuchen"Golden, Bruce"https://zbmath.org/authors/?q=ai:golden.bruce-l"Poikonen, Stefan"https://zbmath.org/authors/?q=ai:poikonen.stefan"Wasil, Edward"https://zbmath.org/authors/?q=ai:wasil.edward-a"Zhang, Rui"https://zbmath.org/authors/?q=ai:zhang.rui.20Summary: Often traditional mail carrier models assume that one carrier is assigned to each truck. This carrier drives the truck according to a delivery route, gets out of the truck at a stop, and services customers using a walking loop. We introduce the Paired Mail Carrier Problem (PMCP), which allows for two mail carriers per truck. If at least one carrier is in the truck, the truck may move forward. Given two mail carriers per truck, the objective is to minimize the time from the start to the end of the route while ensuring that each service stop is fully serviced and obeys all feasibility constraints. We develop a mixed integer programming (MIP) formulation and two fast heuristics for the PMCP. The MIP formulation obtains optimal solutions for smaller instances within reasonable running times. Furthermore, we demonstrate that, in addition to being very efficient (running time is on the order of milliseconds, even for large instances), our heuristics are near-optimal (within 5\% of optimality) on instances up to 80 stops. More importantly, we evaluate the impact of the paired mail carrier (PMC) setting on both a one-truck situation and a fleet (multiple trucks) situation, relative to the single mail carrier (SMC) setting. Overall, the PMC setting not only can accomplish over 50\% extra work within the same shift hours but also can lead to 22\% cost savings. Finally, we discuss an extension where we can have three or more mail carriers per truck.Index-matrix interpretation of a two-stage three-dimensional intuitionistic fuzzy transportation problemhttps://zbmath.org/1522.900062023-12-07T16:00:11.105023Z"Traneva, Velichka"https://zbmath.org/authors/?q=ai:traneva.velichka"Tranev, Stoyan"https://zbmath.org/authors/?q=ai:tranev.stoyanSummary: The transportation problem (TP) is a special type of linear programming problem where the objective is to minimise the cost of distributing a product from a number of sources or origins to a number of destinations. In classical TP, the values of the transportation costs, availability and demand of the products are clear defined. The pandemic situation caused by Covid-19 and rising inflation determine the unclear and rapidly changing values of TP parameters. Uncertain values can be represented by fuzzy sets (FSs), proposed by Zadeh. But there is a more flexible tool for modeling the vague information environment. These are the intuitionistic fuzzy sets (IFSs) proposed by Atanasov, which, in comparison with the fuzzy sets, also have a degree of hesitancy. In this paper we present an index-matrix approach for modeling and solving a two-stage three-dimensional transportation problem (2-S 3-D IFTP), extending the two-stage two-dimensional problem proposed in [\textit{V. Traneva} and \textit{S. Tranev}, ``Two-stage intuitionistic fuzzy transportation problem through the prism of index matrices'', in: Proceedings of the 16th conference on computer science and information systems, FedCSIS 2021. Warsaw: Annals of computer science and information systems 26. 89--96 (2021; \url{doi:10.15439/2021F76})], in which the transportation costs, supply and demand values are intuitionistic fuzzy pairs (IFPs), depending on locations, diesel prices, road condition, weather, time and other factors. Additional constraints are included in the problem: limits for the transportation costs. Its main objective is to determine the quantities of delivery from producers and resselers to buyers to maintain the supply and demand requirements at time (location, etc.) at the cheapest intuitionistic fuzzy transportation cost extending 2-S 2-D IFTP from Traneva and Tranev [loc. cit.]. The solution algorithm is demonstrated by a numerical example.
For the entire collection see [Zbl 1498.90005].Correction to: ``Uncertainty analysis of structural response under a random impact''https://zbmath.org/1522.900142023-12-07T16:00:11.105023Z"Troian, Renata"https://zbmath.org/authors/?q=ai:troian.renata"Lemosse, Didier"https://zbmath.org/authors/?q=ai:lemosse.didier"Khalij, Leila"https://zbmath.org/authors/?q=ai:khalij.leila"de Cursi, Eduardo Souza"https://zbmath.org/authors/?q=ai:souza-de-cursi.jose-eduardoFrom the text: In the original publication [\textit{R. Troian}, ibid. 24, No. 1, 49--64 (2023; Zbl 1519.90064)], authors names were unintentionally confused with the affiliations.
There is an error in the author list of this article, which should be corrected as follows: ``Renata Troian, Didier Lemosse, Leila Khalij, Eduardo Souza de Cursi''.Calibration scheduling with time slot costhttps://zbmath.org/1522.900192023-12-07T16:00:11.105023Z"Wang, Kai"https://zbmath.org/authors/?q=ai:wang.kai.13|wang.kai.4|wang.kai.8|wang.kai.5|wang.kai.6|wang.kai.3|wang.kai.1|wang.kai.2Summary: In this paper we study the scheduling problem with calibration and time slot cost. In this model, the machine has to be calibrated to run a job and the calibration remains valid for a fixed time period of length \(T\), after which it must be recalibrated before running more jobs. On the other hand, a certain cost will be incurred when the machine executes a job and the cost is determined by the time slots occupied by the job in the schedule. We work on the jobs with release times, deadlines and identical processing times. The objective is to schedule the jobs on a single machine and minimize the total cost while calibrating the machine at most \(K\) times. We propose dynamic programmings for different scenarios of this problem, as well as a greedy algorithm for the non-calibration version of this problem.
For the entire collection see [Zbl 1400.68037].Integer interval DEA: an axiomatic derivation of the technology and an additive, slacks-based modelhttps://zbmath.org/1522.900202023-12-07T16:00:11.105023Z"Arana-Jiménez, Manuel"https://zbmath.org/authors/?q=ai:arana-jimenez.manuel"Sánchez-Gil, M. Carmen"https://zbmath.org/authors/?q=ai:sanchez-gil.m-carmen"Younesi, Atefeh"https://zbmath.org/authors/?q=ai:younesi.atefeh"Lozano, Sebastián"https://zbmath.org/authors/?q=ai:lozano.sebastianSummary: The present paper studies the efficiency assessment of Decision-Making Units (DMUs) when their inputs and outputs are described under uncertainty of the type of integer interval data. An axiomatic derivation of the production possibility set (PPS) is presented. An additive, slacks-based data envelopment analysis (DEA) model is formulated, consisting of two phases. This has required the use of adequate arithmetic and LU-partial orders for integer intervals. This novel integer interval DEA approach is the first step towards DEA models under fuzzy integer intervals, with the extension of the corresponding arithmetic and LU-partial orders to fuzzy integer intervals. The proposed method is applied on a dataset, taken from the literature, that involves both continuous and integer interval variables.Proportionate change of outputs and of inputs in DEAhttps://zbmath.org/1522.900212023-12-07T16:00:11.105023Z"Neralić, Luka"https://zbmath.org/authors/?q=ai:neralic.lukaSummary: In this review paper, we consider sensitivity analysis for the proportionate change of outputs and of inputs in data envelopment analysis (DEA). Sensitivity analysis is studied in the additive model for the proportionate changes of outputs, of inputs, and of simultaneous proportionate changes of outputs and of inputs. Cases of proportionate changes of outputs and inputs with different coefficients of proportionality and of changes of discretionary outputs and inputs are also studied for the additive model. Sensitivity analysis in the CCR model is considered for the simultaneous proportionate changes of outputs and of inputs, for the proportionate changes of a subset of inputs and/or of outputs and for the proportionate changes of outputs and inputs with different coefficients of proportionality.The \(k\)-power domination problem in weighted treeshttps://zbmath.org/1522.900222023-12-07T16:00:11.105023Z"Cheng, ChangJie"https://zbmath.org/authors/?q=ai:cheng.changjie"Lu, Changhong"https://zbmath.org/authors/?q=ai:lu.changhong"Zhou, Yu"https://zbmath.org/authors/?q=ai:zhou.yu.3Summary: The power domination problem of the graph comes from how to choose the node location problem of the least phase measurement units in the electric power system. In the actual electric power system, because of the difference in the cost of phase measurement units at different nodes, it is more practical to study the power domination problem with the weighted graph. In this paper, we present a dynamic programming style linear-time algorithm for \(k\)-power domination problem in weighted trees.
For the entire collection see [Zbl 1400.68037].Research of production groups formation problem subject to logical restrictionshttps://zbmath.org/1522.900232023-12-07T16:00:11.105023Z"Afanasyeva, Lubov D."https://zbmath.org/authors/?q=ai:afanasyeva.lubov-d"Kolokolov, Alexander A."https://zbmath.org/authors/?q=ai:kolokolov.aleksandr-aleksandrovichSummary: This paper is devoted to the production groups formation problem subject to logical restrictions, reflecting interpersonal relations in a team. Mathematical models are developed and investigated using graph theory and linear integer programming, a number of algorithms of combinatorial type is presented, their theoretical and experimental analyses are conducted.Faster first-order primal-dual methods for linear programming using restarts and sharpnesshttps://zbmath.org/1522.900242023-12-07T16:00:11.105023Z"Applegate, David"https://zbmath.org/authors/?q=ai:applegate.david-l|applegate.david-a"Hinder, Oliver"https://zbmath.org/authors/?q=ai:hinder.oliver"Lu, Haihao"https://zbmath.org/authors/?q=ai:lu.haihao"Lubin, Miles"https://zbmath.org/authors/?q=ai:lubin.milesSummary: First-order primal-dual methods are appealing for their low memory overhead, fast iterations, and effective parallelization. However, they are often slow at finding high accuracy solutions, which creates a barrier to their use in traditional linear programming (LP) applications. This paper exploits the sharpness of primal-dual formulations of LP instances to achieve linear convergence using restarts in a general setting that applies to alternating direction method of multipliers (ADMM), primal-dual hybrid gradient method (PDHG) and extragradient method (EGM). In the special case of PDHG, without restarts we show an iteration count lower bound of \(\Omega (\kappa^2 \log (1/\epsilon))\), while with restarts we show an iteration count upper bound of \(O(\kappa \log (1/\epsilon))\), where \(\kappa\) is a condition number and \(\epsilon\) is the desired accuracy. Moreover, the upper bound is optimal for a wide class of primal-dual methods, and applies to the strictly more general class of sharp primal-dual problems. We develop an adaptive restart scheme and verify that restarts significantly improve the ability of PDHG, EGM, and ADMM to find high accuracy solutions to LP problems.The polyhedral geometry of pivot rules and monotone pathshttps://zbmath.org/1522.900252023-12-07T16:00:11.105023Z"Black, Alexander E."https://zbmath.org/authors/?q=ai:black.alexander-e"De Loera, Jesús A."https://zbmath.org/authors/?q=ai:de-loera.jesus-a"Lütjeharms, Niklas"https://zbmath.org/authors/?q=ai:lutjeharms.niklas"Sanyal, Raman"https://zbmath.org/authors/?q=ai:sanyal.ramanSummary: Motivated by the analysis of the performance of the simplex method, we study the behavior of families of pivot rules of linear programs. We introduce \textit{normalized-weight pivot rules} which are fundamental for the following reasons: First, they are \textit{memory-less,} in the sense that the pivots are governed by local information encoded by an arborescence. Second, many of the most used pivot rules belong to that class, and we show this subclass is critical for understanding the complexity of all pivot rules. Finally, normalized-weight pivot rules can be parametrized in a natural continuous manner. The latter leads to the existence of two polytopes, the \textit{pivot rule polytopes} and the \textit{neighbotopes,} that capture the behavior of normalized-weight pivot rules on polytopes and linear programs. We explain their face structure in terms of multi-arborescences and compute upper bounds on the number of coherent arborescences, that is, vertices of our polytopes. We introduce a normalized-weight pivot rule, called the \textit{max-slope pivot rule}, which generalizes the shadow-vertex pivot rule. The corresponding pivot rule polytopes and neighbotopes refine the \textit{monotone path polytopes} of Billera and Sturmfels. Our constructions are important beyond optimization and provide new perspectives on classical geometric combinatorics. Special cases of our polytopes yield permutahedra, associahedra, and multiplihedra. For the greatest-improvement pivot rules we draw connections to sweep polytopes and polymatroids.Efficient joint object matching via linear programminghttps://zbmath.org/1522.900262023-12-07T16:00:11.105023Z"De Rosa, Antonio"https://zbmath.org/authors/?q=ai:de-rosa.antonio"Khajavirad, Aida"https://zbmath.org/authors/?q=ai:khajavirad.aidaSummary: Joint object matching, also known as multi-image matching, namely, the problem of finding consistent partial maps among all pairs of objects within a collection, is a crucial task in many areas of computer vision. This problem subsumes bipartite graph matching and graph partitioning as special cases and is NP-hard, in general. We develop scalable linear programming (LP) relaxations with theoretical performance guarantees for joint object matching. We start by proposing a new characterization of consistent partial maps; this in turn enables us to formulate joint object matching as an integer linear programming (ILP) problem. To construct strong LP relaxations, we study the facial structure of the convex hull of the feasible region of this ILP, which we refer to as the joint matching polytope. We present an exponential family of facet-defining inequalities that can be separated in strongly polynomial time, hence obtaining a partial characterization of the joint matching polytope that is both tight and cheap to compute. To analyze the theoretical performance of the proposed LP relaxations, we focus on permutation group synchronization, an important special case of joint object matching. We show that under the random corruption model for the input maps, a simple LP relaxation, that is, an LP containing only a very small fraction of the proposed facet-defining inequalities, recovers the ground truth with high probability if the corruption level is below 40\%. Finally, via a preliminary computational study on synthetic data, we show that the proposed LP relaxations outperform a popular SDP relaxation both in terms of recovery and tightness.Current limit avoidance algorithms for DEMO operationhttps://zbmath.org/1522.900272023-12-07T16:00:11.105023Z"di Grazia, Luigi Emanuel"https://zbmath.org/authors/?q=ai:di-grazia.luigi-emanuel"Frattolillo, Domenico"https://zbmath.org/authors/?q=ai:frattolillo.domenico"De Tommasi, Gianmaria"https://zbmath.org/authors/?q=ai:de-tommasi.gianmaria"Mattei, Massimiliano"https://zbmath.org/authors/?q=ai:mattei.massimilianoSummary: Tokamaks are the most promising devices to prove the feasibility of energy production using nuclear fusion on Earth which is foreseen as a possible source of energy for the next centuries. In large tokamaks with superconducting poloidal field (PF) coils, the problem of avoiding saturation of the currents is of paramount importance, especially for a reactor such as the European demonstration fusion power plant DEMO. Indeed, reaching the current limits during plasma operation may cause a loss of control of the plasma shape and/or current, leading to a major disruption. Therefore, a current limit avoidance (CLA) system is essential to assure safe operation. Three different algorithms to be implemented within a CLA system are proposed in this paper: two are based on online solutions of constrained optimization problems, while the third one relies on dynamic allocation. The performance assessment for all the proposed solutions is carried out by considering challenging operation scenarios for the DEMO reactor, such as the case where more than one PF current simultaneously saturates during the discharge. An evaluation of the computational burden needed to solve the allocation problem for the various proposed alternatives is also presented, which shows the compliance of the optimization-based approaches with the envisaged deadlines for real-time implementation of the DEMO plasma magnetic control system.A full-Newton step infeasible interior-point algorithm based on a kernel function with a new barrier termhttps://zbmath.org/1522.900282023-12-07T16:00:11.105023Z"Guerdouh, Safa"https://zbmath.org/authors/?q=ai:guerdouh.safa"Chikouche, Wided"https://zbmath.org/authors/?q=ai:chikouche.wided"Kheirfam, Behrouz"https://zbmath.org/authors/?q=ai:kheirfam.behrouz(no abstract)Centering ADMM for the semidefinite relaxation of the QAPhttps://zbmath.org/1522.900292023-12-07T16:00:11.105023Z"Kanoh, Shin-ichi"https://zbmath.org/authors/?q=ai:kanoh.shin-ichi"Yoshise, Akiko"https://zbmath.org/authors/?q=ai:yoshise.akikoSummary: We propose a new method for solving the semidefinite (SD) relaxation of the quadratic assignment problem (QAP), called centering ADMM. Centering ADMM is an alternating direction method of multipliers (ADMM) combining the centering steps used in the interior-point method. The first stage of centering ADMM updates the iterate so that it approaches the central path by incorporating a barrier function term into the objective function, as in the interior-point method. If the current iterate is sufficiently close to the central path with a sufficiently small value of the barrier parameter, the method switches to the standard version of ADMM. We show that centering ADMM (not employing a dynamic update of the penalty parameter) has global convergence properties. To observe the effect of the centering steps, we conducted numerical experiments with SD relaxation problems of instances in QAPLIB. The results demonstrate that the centering steps are quite efficient for some classes of instances.
For the entire collection see [Zbl 1480.00062].Row-oriented decomposition in large-scale linear optimizationhttps://zbmath.org/1522.900302023-12-07T16:00:11.105023Z"Nurminski, Evgeni"https://zbmath.org/authors/?q=ai:nurminski.evgeni-a"Shamray, Natalia"https://zbmath.org/authors/?q=ai:shamray.natalia-bSummary: The single-projection linear optimization method [\textit{E. A. Nurminski}, J. Glob. Optim. 66, No. 1, 95--110 (2016; Zbl 1349.90608)] demonstrated a promising computational performance on the series of giga-scale academic and practical problems shown in this talk. Another attractive feature of this method is its potential for row-wise decomposition of large-scale problems. It can be applied irrelevant to the problem structure but also can make use of it if present. This decomposition technique might take different forms as well, and a few variants will be presented.
For the entire collection see [Zbl 1508.90001].A dual-based stochastic inexact algorithm for a class of stochastic nonsmooth convex composite problemshttps://zbmath.org/1522.900312023-12-07T16:00:11.105023Z"Lin, Gui-Hua"https://zbmath.org/authors/?q=ai:lin.guihua"Yang, Zhen-Ping"https://zbmath.org/authors/?q=ai:yang.zhenping"Yin, Hai-An"https://zbmath.org/authors/?q=ai:yin.hai-an"Zhang, Jin"https://zbmath.org/authors/?q=ai:zhang.jin.2Summary: In this paper, a dual-based stochastic inexact algorithm is developed to solve a class of stochastic nonsmooth convex problems with underlying structure. This algorithm can be regarded as an integration of a deterministic augmented Lagrangian method and some stochastic approximation techniques. By utilizing the sparsity of the second order information, each subproblem is efficiently solved by a superlinearly convergent semismooth Newton method. We derive some almost surely convergence properties and convergence rate of objective values. Furthermore, we present some results related to convergence rate of distance between iteration points and solution set under error bound conditions. Numerical results demonstrate favorable comparison of the proposed algorithm with some existing methods.A decomposition augmented Lagrangian method for low-rank semidefinite programminghttps://zbmath.org/1522.900322023-12-07T16:00:11.105023Z"Wang, Yifei"https://zbmath.org/authors/?q=ai:wang.yifei"Deng, Kangkang"https://zbmath.org/authors/?q=ai:deng.kangkang"Liu, Haoyang"https://zbmath.org/authors/?q=ai:liu.haoyang"Wen, Zaiwen"https://zbmath.org/authors/?q=ai:wen.zaiwenSummary: We develop a decomposition method based on the augmented Lagrangian framework to solve a broad family of semidefinite programming problems, possibly with nonlinear objective functions, nonsmooth regularization, and general linear equality/inequality constraints. In particular, the positive semidefinite variable along with a group of linear constraints can be transformed into a variable on a smooth manifold via matrix factorization. The nonsmooth regularization and other general linear constraints are handled by the augmented Lagrangian method. Therefore, each subproblem can be solved by a semismooth Newton method on a manifold. Theoretically, we show that the first and second-order necessary optimality conditions for the factorized subproblem are also sufficient for the original subproblem under certain conditions. Convergence analysis is established for the Riemannian subproblem and the augmented Lagrangian method. Extensive numerical experiments on large-scale semidefinite programming problems such as max-cut, nearest correlation estimation, clustering, and sparse principal component analysis demonstrate the strength of our proposed method compared to other state-of-the-art methods.An inexact projected gradient method with rounding and lifting by nonlinear programming for solving rank-one semidefinite relaxation of polynomial optimizationhttps://zbmath.org/1522.900332023-12-07T16:00:11.105023Z"Yang, Heng"https://zbmath.org/authors/?q=ai:yang.heng"Liang, Ling"https://zbmath.org/authors/?q=ai:liang.ling"Carlone, Luca"https://zbmath.org/authors/?q=ai:carlone.luca"Toh, Kim-Chuan"https://zbmath.org/authors/?q=ai:toh.kimchuanSummary: We consider solving high-order and tight semidefinite programming (SDP) relaxations of nonconvex polynomial optimization problems (POPs) that often admit degenerate rank-one optimal solutions. Instead of solving the SDP alone, we propose a new algorithmic framework that blends \textit{local search} using the nonconvex POP into \textit{global descent} using the convex SDP. In particular, we first design a globally convergent \textit{inexact} projected gradient method (iPGM) for solving the SDP that serves as the backbone of our framework. We then accelerate iPGM by taking long, but \textit{safeguarded}, rank-one steps generated by fast nonlinear programming algorithms. We prove that the new framework is still globally convergent for solving the SDP. To solve the iPGM subproblem of projecting a given point onto the feasible set of the SDP, we design a two-phase algorithm with phase one using a symmetric Gauss-Seidel based accelerated proximal gradient method (sGS-APG) to generate a good initial point, and phase two using a modified limited-memory BFGS (L-BFGS) method to obtain an accurate solution. We analyze the convergence for both phases and establish a novel global convergence result for the modified L-BFGS that does not require the objective function to be twice continuously differentiable. We conduct numerical experiments for solving second-order SDP relaxations arising from a diverse set of POPs. Our framework demonstrates state-of-the-art efficiency, scalability, and robustness in solving degenerate SDPs to high accuracy, even in the presence of millions of equality constraints.A class of fast iterative shrinkage thresholding algorithm for nonsmooth optimizationhttps://zbmath.org/1522.900342023-12-07T16:00:11.105023Z"Zhou, Hanlin"https://zbmath.org/authors/?q=ai:zhou.hanlin"Cheng, Wanyou"https://zbmath.org/authors/?q=ai:cheng.wanyou"Ye, Jianhao"https://zbmath.org/authors/?q=ai:ye.jianhao"Zhang, Jiahao"https://zbmath.org/authors/?q=ai:zhang.jiahaoSummary: We propose a class of fast iterative shrinkage thresholding algorithm which contains the original FISTA scheme, the FISTA-CD scheme and the FISTA-Mod. We prove that the objective value sequence achieves \(O(\frac 1{k^2})\) complexity. Moreover, we prove that any limit point of the sequence generated by the proposed algorithm is a minimizer of the objective function. At last, we propose a modified adaptive restart strategy which can dramatically improve the convergence rate of the proposed algorithms. Numerical results demonstrate that the proposed algorithm is competitive with several known methods.Maintenance groups evaluation under uncertainties: a novel stochastic free disposal hull in the presence of lognormally distributed datahttps://zbmath.org/1522.900352023-12-07T16:00:11.105023Z"Dibachi, Hossein"https://zbmath.org/authors/?q=ai:dibachi.hossein"Izadikhah, Mohammad"https://zbmath.org/authors/?q=ai:izadikhah.mohammadSummary: Maintenance groups play an essential role in the successful operation of large companies and factories. Additionally, data envelopment analysis (DEA) is known as a valuable tool for monitoring the performance of maintenance groups. Especially, in contrast to the conventional DEA models that impose the convexity assumption into the technology, the free disposal hull (FDH) model provides a method for assessing the efficiency without the assumption of convexity and can be considered a valuable tool for determining one of the observed groups as the benchmark for each maintenance group. Meanwhile, because of the stochastic structure of data with lognormal distribution in the maintenance groups, this paper extends the FDH model in stochastic data with the lognormal distribution. Moreover, the method's capabilities are confirmed based on some theorems, and a simulation study that illustrated the properties of the developed procedure is also performed. The developed methodology is applied to assess the performance of 21 maintenance groups of AZCO under uncertainty conditions.Mixed-integer programming techniques for the minimum sum-of-squares clustering problemhttps://zbmath.org/1522.900362023-12-07T16:00:11.105023Z"Burgard, Jan Pablo"https://zbmath.org/authors/?q=ai:burgard.jan-pablo"Costa, Carina Moreira"https://zbmath.org/authors/?q=ai:costa.carina-moreira"Hojny, Christopher"https://zbmath.org/authors/?q=ai:hojny.christopher"Kleinert, Thomas"https://zbmath.org/authors/?q=ai:kleinert.thomas"Schmidt, Martin"https://zbmath.org/authors/?q=ai:schmidt.martinSummary: The minimum sum-of-squares clustering problem is a very important problem in data mining and machine learning with very many applications in, e.g., medicine or social sciences. However, it is known to be NP-hard in all relevant cases and to be notoriously hard to be solved to global optimality in practice. In this paper, we develop and test different tailored mixed-integer programming techniques to improve the performance of state-of-the-art MINLP solvers when applied to the problem -- among them are cutting planes, propagation techniques, branching rules, or primal heuristics. Our extensive numerical study shows that our techniques significantly improve the performance of the open-source MINLP solver SCIP. Consequently, using our novel techniques, we can solve many instances that are not solvable with SCIP without our techniques and we obtain much smaller gaps for those instances that can still not be solved to global optimality.Scanning integer points with lex-inequalities: a finite cutting plane algorithm for integer programming with linear objectivehttps://zbmath.org/1522.900372023-12-07T16:00:11.105023Z"Conforti, Michele"https://zbmath.org/authors/?q=ai:conforti.michele"De Santis, Marianna"https://zbmath.org/authors/?q=ai:de-santis.marianna"Di Summa, Marco"https://zbmath.org/authors/?q=ai:di-summa.marco"Rinaldi, Francesco"https://zbmath.org/authors/?q=ai:rinaldi.francescoSummary: We consider the integer points in a unimodular cone \(K\) ordered by a lexicographic rule defined by a lattice basis. To each integer point \(x\) in \(K\) we associate a family of inequalities (lex-inequalities) that define the convex hull of the integer points in \(K\) that are not lexicographically smaller than \(x\). The family of lex-inequalities contains the Chvátal-Gomory cuts, but does not contain and is not contained in the family of split cuts. This provides a finite cutting plane method to solve the integer program \(\min \{cx: x\in S\cap \mathbb{Z}^n\} \), where \(S\subset \mathbb{R}^n\) is a compact set and \(c\in \mathbb{Z}^n\). We analyze the number of iterations of our algorithm.A novel reformulation for the single-sink fixed-charge transportation problemhttps://zbmath.org/1522.900382023-12-07T16:00:11.105023Z"Legault, Robin"https://zbmath.org/authors/?q=ai:legault.robin"Côté, Jean-François"https://zbmath.org/authors/?q=ai:cote.jean-francois"Gendron, Bernard"https://zbmath.org/authors/?q=ai:gendron.bernardSummary: The single-sink fixed-charge transportation problem is known to have many applications in the area of manufacturing and transportation as well as being an important subproblem of the fixed-charge transportation problem. However, even the best algorithms from the literature do not fully leverage the structure of this problem, to the point of being surpassed by modern general-purpose mixed-integer programming solvers for large instances. We introduce a novel reformulation of the problem and study its theoretical properties. This reformulation leads to a range of new upper and lower bounds, dominance relations, linear relaxations, and filtering procedures. The resulting algorithm includes a heuristic phase and an exact phase, the main step of which is to solve a very small number of knapsack subproblems. Computational experiments are presented for existing and new types of instances. These tests indicate that the new algorithm systematically reduces the resolution time of the state-of-the-art exact methods by several orders of magnitude.Algorithms for the genome median under a restricted measure of rearrangementhttps://zbmath.org/1522.900392023-12-07T16:00:11.105023Z"Silva, Helmuth O. M."https://zbmath.org/authors/?q=ai:silva.helmuth-o-m"Rubert, Diego P."https://zbmath.org/authors/?q=ai:rubert.diego-p"Araujo, Eloi"https://zbmath.org/authors/?q=ai:araujo.eloi"Steffen, Eckhard"https://zbmath.org/authors/?q=ai:steffen.eckhard"Doerr, Daniel"https://zbmath.org/authors/?q=ai:doerr.daniel"Martinez, Fábio V."https://zbmath.org/authors/?q=ai:martinez.fabio-viduaniSummary: Ancestral reconstruction is a classic task in comparative genomics. Here, we study the \textit{genome median problem}, a related computational problem which, given a set of three or more genomes, asks to find a new genome that minimizes the sum of pairwise distances between it and the given genomes. The \textit{distance} stands for the amount of evolution observed at the genome level, for which we determine the minimum number of rearrangement operations necessary to transform one genome into the other. For almost all rearrangement operations the median problem is NP-hard, with the exception of the \textit{breakpoint median} that can be constructed efficiently for multichromosomal circular and mixed genomes. In this work, we study the median problem under a restricted rearrangement measure called \(c_4\)-\textit{distance}, which is closely related to the breakpoint and the DCJ distance. We identify tight bounds and decomposers of the \(c_4\)-median and develop algorithms for its construction, one exact ILP-based and three combinatorial heuristics. Subsequently, we perform experiments on simulated data sets. Our results suggest that the \(c_4\)-distance is useful for the study the genome median problem, from theoretical and practical perspectives.Strong valid inequalities for a class of concave submodular minimization problems under cardinality constraintshttps://zbmath.org/1522.900402023-12-07T16:00:11.105023Z"Yu, Qimeng"https://zbmath.org/authors/?q=ai:yu.qimeng"Küçükyavuz, Simge"https://zbmath.org/authors/?q=ai:kucukyavuz.simgeSummary: We study the polyhedral convex hull structure of a mixed-integer set which arises in a class of cardinality-constrained concave submodular minimization problems. This class of problems has an objective function in the form of \(f(a^\top x)\), where \(f\) is a univariate concave function, \(a\) is a non-negative vector, and \(x\) is a binary vector of appropriate dimension. Such minimization problems frequently appear in applications that involve risk-aversion or economies of scale. We propose three classes of strong valid linear inequalities for this convex hull and specify their facet conditions when \(a\) has two distinct values. We show how to use these inequalities to obtain valid inequalities for general \(a\) that contains multiple values. We further provide a complete linear convex hull description for this mixed-integer set when \(a\) contains two distinct values and the cardinality constraint upper bound is two. Our computational experiments on the mean-risk optimization problem demonstrate the effectiveness of the proposed inequalities in a branch-and-cut framework.Supermodularity and valid inequalities for quadratic optimization with indicatorshttps://zbmath.org/1522.900412023-12-07T16:00:11.105023Z"Atamtürk, Alper"https://zbmath.org/authors/?q=ai:atamturk.alper"Gómez, Andrés"https://zbmath.org/authors/?q=ai:gomez.andresSummary: We study the minimization of a rank-one quadratic with indicators and show that the underlying set function obtained by projecting out the continuous variables is supermodular. Although supermodular minimization is, in general, difficult, the specific set function for the rank-one quadratic can be minimized in linear time. We show that the convex hull of the epigraph of the quadratic can be obtained from inequalities for the underlying supermodular set function by lifting them into nonlinear inequalities in the original space of variables. Explicit forms of the convex-hull description are given, both in the original space of variables and in an extended formulation via conic quadratic-representable inequalities, along with a polynomial separation algorithm. Computational experiments indicate that the lifted supermodular inequalities in conic quadratic form are quite effective in reducing the integrality gap for quadratic optimization with indicators.Properties, extensions and application of piecewise linearization for Euclidean norm optimization in \(\mathbb{R}^2\)https://zbmath.org/1522.900422023-12-07T16:00:11.105023Z"Duguet, Aloïs"https://zbmath.org/authors/?q=ai:duguet.alois"Artigues, Christian"https://zbmath.org/authors/?q=ai:artigues.christian"Houssin, Laurent"https://zbmath.org/authors/?q=ai:houssin.laurent"Ngueveu, Sandra Ulrich"https://zbmath.org/authors/?q=ai:ngueveu.sandra-ulrichSummary: This work considers nonconvex mixed integer nonlinear programming where nonlinearity comes from the presence of the two-dimensional euclidean norm in the objective or the constraints. We build from the euclidean norm piecewise linearization proposed by \textit{J.-T. Camino} et al. [Comput. Optim. Appl. 73, No. 2, 679--705 (2019; Zbl 1423.90152)] that allows to solve such nonconvex problems via mixed-integer linear programming with an arbitrary approximation guarantee. Theoretical results are established that prove that this linearization is able to satisfy any given approximation level with the minimum number of pieces. An extension of the piecewise linearization approach is proposed. It shares the same theoretical properties for elliptic constraints and/or objective. An application shows the practical appeal of the elliptic linearization on a nonconvex beam layout mixed optimization problem coming from an industrial application.\(2 \times 2\)-convexifications for convex quadratic optimization with indicator variableshttps://zbmath.org/1522.900432023-12-07T16:00:11.105023Z"Han, Shaoning"https://zbmath.org/authors/?q=ai:han.shaoning"Gómez, Andrés"https://zbmath.org/authors/?q=ai:gomez.andres"Atamtürk, Alper"https://zbmath.org/authors/?q=ai:atamturk.alperSummary: In this paper, we study the convex quadratic optimization problem with indicator variables. For the \(2 \times 2\) case, we describe the convex hull of the epigraph in the original space of variables, and also give a conic quadratic extended formulation. Then, using the convex hull description for the \(2 \times 2\) case as a building block, we derive an extended SDP relaxation for the general case. This new formulation is stronger than other SDP relaxations proposed in the literature for the problem, including the optimal perspective relaxation and the optimal rank-one relaxation. Computational experiments indicate that the proposed formulations are quite effective in reducing the integrality gap of the optimization problems.An approximation algorithm for indefinite mixed integer quadratic programminghttps://zbmath.org/1522.900442023-12-07T16:00:11.105023Z"Pia, Alberto Del"https://zbmath.org/authors/?q=ai:del-pia.albertoSummary: In this paper, we give an algorithm that finds an \(\epsilon\)-approximate solution to a mixed integer quadratic programming (MIQP) problem. The algorithm runs in polynomial time if the rank of the quadratic function and the number of integer variables are fixed. The running time of the algorithm is expected unless P = NP. In order to design this algorithm we introduce the novel concepts of spherical form MIQP and of aligned vectors, and we provide a number of results of independent interest. In particular, we give a strongly polynomial algorithm to find a symmetric decomposition of a matrix, and show a related result on simultaneous diagonalization of matrices.Adaptive cut selection in mixed-integer linear programminghttps://zbmath.org/1522.900452023-12-07T16:00:11.105023Z"Turner, Mark"https://zbmath.org/authors/?q=ai:turner.mark-g"Koch, Thorsten"https://zbmath.org/authors/?q=ai:koch.thorsten"Serrano, Felipe"https://zbmath.org/authors/?q=ai:serrano.felipe"Winkler, Michael"https://zbmath.org/authors/?q=ai:winkler.michaelSummary: Cutting plane selection is a subroutine used in all modern mixed-integer linear programming solvers with the goal of selecting a subset of generated cuts that induce optimal solver performance. These solvers have millions of parameter combinations, and so are excellent candidates for parameter tuning. Cut selection scoring rules are usually weighted sums of different measurements, where the weights are parameters. We present a parametric family of mixed-integer linear programs together with infinitely many family-wide valid cuts. Some of these cuts can induce integer optimal solutions directly after being applied, while others fail to do so even if an infinite amount are applied. We show for a specific cut selection rule, that any finite grid search of the parameter space will always miss all parameter values, which select integer optimal inducing cuts in an infinite amount of our problems. We propose a variation on the design of existing graph convolutional neural networks, adapting them to learn cut selection rule parameters. We present a reinforcement learning framework for selecting cuts, and train our design using said framework over MIPLIB 2017 and a neural network verification data set. Our framework and design show that adaptive cut selection does substantially improve performance over a diverse set of instances, but that finding a single function describing such a rule is difficult. Code for reproducing all experiments is available at \url{https://github.com/Opt-Mucca/Adaptive-Cutsel-MILP}.Time block decomposition of multistage stochastic optimization problemshttps://zbmath.org/1522.900462023-12-07T16:00:11.105023Z"Carpentier, Pierre"https://zbmath.org/authors/?q=ai:carpentier.pierre"Chancelier, Jean-Philippe"https://zbmath.org/authors/?q=ai:chancelier.jean-philippe"De Lara, Michel"https://zbmath.org/authors/?q=ai:de-lara.michel"Martin, Thomas"https://zbmath.org/authors/?q=ai:martin.thomas-j|martin.thomas-g"Rigaut, Tristan"https://zbmath.org/authors/?q=ai:rigaut.tristanSummary: Multistage stochastic optimization problems are, by essence, complex as their solutions are functions of both stages and uncertainties. Their large scale nature makes decomposition methods appealing, like dynamic programming which is a sequential decomposition using a state variable defined at all stages. By contrast, in this paper we introduce the notion of state reduction by time blocks, that is, at stages that are not necessarily all the original stages. Then, we prove a dynamic programming equation with value functions that are functions of a state only at some stages. This equation crosses over time blocks, but involves a dynamic optimization inside each block. We illustrate our contribution by showing its potential in three applications in multistage stochastic optimization: mixing dynamic programming and stochastic programming, two-time-scale optimization problems, decision-hazard-decision optimization problems.Dynamic programming for data independent decision setshttps://zbmath.org/1522.900472023-12-07T16:00:11.105023Z"Dommel, Paul"https://zbmath.org/authors/?q=ai:dommel.paul"Pichler, Alois"https://zbmath.org/authors/?q=ai:pichler.aloisSummary: Multistage stochastic optimization problems are oftentimes formulated informally in a pathwise way. These formulations are appealing in a discrete setting and suitable when addressing computational challenges, for example. But the pathwise problem statement does not allow an analysis with mathematical rigor and is therefore not appropriate.
\textit{R. T. Rockafellar} and \textit{R. J. B. Wets} [in: Stoch. Syst.: Model., Identif., Optim. II; Math. Program. Study 6, 170--187 (1976; Zbl 0377.90073)] address the fundamental measurability concern of the value functions in the case of convex costs and constraints. This paper resumes these foundations. The contribution is a proof that there exist measurable versions of intermediate value functions, which reveals regularity in addition. Our proof builds on the Kolmogorov continuity theorem.
It is demonstrated that verification theorems allow stating traditional problem specifications in the novel setting with mathematical rigor. Further, we provide dynamic equations for the general problem. The problem classes covered include Markov decision processes, reinforcement learning and stochastic dual dynamic programming.Epi-convergence of expectation functions under varying measures and integrandshttps://zbmath.org/1522.900482023-12-07T16:00:11.105023Z"Feinberg, Eugene A."https://zbmath.org/authors/?q=ai:feinberg.eugene-a"Kasyanov, Pavlo O."https://zbmath.org/authors/?q=ai:kasyanov.pavlo-o"Royset, Johannes O."https://zbmath.org/authors/?q=ai:royset.johannes-oSummary: For expectation functions on metric spaces, we provide sufficient conditions for epi-convergence under varying probability measures and integrands, and examine applications in the area of sieve estimators, mollifier smoothing, PDE-constrained optimization, and stochastic optimization with expectation constraints. As a stepping stone to epi-convergence of independent interest, we develop parametric Fatou's lemmas under mild integrability assumptions. In the setting of Suslin metric spaces, the assumptions are expressed in terms of Pasch-Hausdorff envelopes. For general metric spaces, the assumptions shift to semicontinuity of integrands also on the sample space, which then is assumed to be a metric space.Trajectory following dynamic programming algorithms without finite support assumptionshttps://zbmath.org/1522.900492023-12-07T16:00:11.105023Z"Forcier, Maël"https://zbmath.org/authors/?q=ai:forcier.mael"Leclère, Vincent"https://zbmath.org/authors/?q=ai:leclere.vincentSummary: We introduce a class of algorithms, called Trajectory Following Dynamic Programming (TFDP) algorithms, that iteratively refines approximations of cost-to-go functions of multistage stochastic problems with independent random variables. This framework encompasses most variants of the Stochastic Dual Dynamic Programming algorithm. Leveraging a Lipschitz assumption on the expected cost-to-go functions, we provide a new convergence and complexity proof that allows random variables with non-finitely supported distributions. In particular, this leads to new complexity results for numerous known algorithms. Further, we detail how TFDP algorithms can be implemented without the finite support assumption, either through approximations or exact computations.Data perturbations in stochastic generalized equations: statistical robustness in static and sample average approximated modelshttps://zbmath.org/1522.900502023-12-07T16:00:11.105023Z"Guo, Shaoyan"https://zbmath.org/authors/?q=ai:guo.shaoyan"Xu, Huifu"https://zbmath.org/authors/?q=ai:xu.huifuSummary: Sample average approximation which is also known as Monte Carlo method has been widely used for solving stochastic programming and equilibrium problems. In a data-driven environment, samples are often drawn from empirical data and hence may be potentially contaminated. Consequently it is legitimate to ask whether statistical estimators obtained from solving the sample average approximated problems are statistically robust, that is, the difference between the laws of the statistical estimators based on contaminated data and real data is controllable under some metrics. In Guo and Xu (Math Program 190:679-720, 2021), we address the issue for the estimators of the optimal values of a wide range of stochastic programming problems. In this paper, we complement the research by investigating the optimal solution estimators and we do so by considering stochastic generalized equations (SGE) as a unified framework. Specifically, we look into the impact of a single data perturbation on the solutions of the SGE using the notion of influence function in robust statistics. Since the SGE may have multiple solutions, we use the proto-derivative of a set-valued mapping to introduce the notion of generalized influence function (GIF) and derive sufficient conditions under which the GIF is well defined, bounded and uniformly bounded. We then move on to quantitative statistical analysis of the SGE when all of sample data are potentially contaminated and demonstrate under moderate conditions quantitative statistical robustness of the solutions obtained from solving sample average approximated SGE.Bayesian joint chance constrained optimization: approximations and statistical consistencyhttps://zbmath.org/1522.900512023-12-07T16:00:11.105023Z"Jaiswal, Prateek"https://zbmath.org/authors/?q=ai:jaiswal.prateek"Honnappa, Harsha"https://zbmath.org/authors/?q=ai:honnappa.harsha"Rao, Vinayak A."https://zbmath.org/authors/?q=ai:rao.vinayak-aSummary: This paper considers data-driven chance-constrained stochastic optimization problems in a Bayesian framework. Bayesian posteriors afford a principled mechanism to incorporate data and prior knowledge into stochastic optimization problems. However, the computation of Bayesian posteriors is typically an intractable problem and has spawned a large literature on approximate Bayesian computation. Here, in the context of chance-constrained optimization, we focus on the question of statistical consistency (in an appropriate sense) of the optimal value, computed using an approximate posterior distribution. To this end, we rigorously prove a frequentist consistency result demonstrating the convergence of the optimal value to the optimal value of a fixed, parameterized constrained optimization problem. We augment this by also establishing a probabilistic rate of convergence of the optimal value. We also prove the convex feasibility of the approximate Bayesian stochastic optimization problem. Finally, we demonstrate the utility of our approach on an optimal staffing problem for an \(M/M/c\) queueing model.A smoothing projected HS method for solving stochastic tensor complementarity problemhttps://zbmath.org/1522.900522023-12-07T16:00:11.105023Z"Lu, Mengdie"https://zbmath.org/authors/?q=ai:lu.mengdie"Du, Shouqiang"https://zbmath.org/authors/?q=ai:du.shouqiangSummary: The stochastic tensor complementarity problem is a class of optimization problem with a wide application future. We reformulate the stochastic tensor complementarity problem as an expected residual minimization problem which aims to minimize an expected residual function defined by the restricted nonlinear complementarity function. Combined with the projected gradient method which has simple structure and low computational cost, we propose a new smoothing projected Hesteness-Stiefel (HS) method to solve the stochastic tensor complementarity problem. We establish the global convergence of the proposed method. Relevant numerical results compared with the traditional projected gradient method are also given to show the efficiency of the proposed method.Dynamic programming in convex stochastic optimizationhttps://zbmath.org/1522.900532023-12-07T16:00:11.105023Z"Pennanen, Teemu"https://zbmath.org/authors/?q=ai:pennanen.teemu"Perkkiö, Ari-Pekka"https://zbmath.org/authors/?q=ai:perkkio.ari-pekkaSummary: This paper studies the dynamic programming principle for general convex stochastic optimization problems introduced by \textit{R. T. Rockafellar} and \textit{R. J. B. Wets} [in: Stoch. Syst.: Model., Identif., Optim. II; Math. Program. Study 6, 170--187 (1976; Zbl 0377.90073)]. We extend the applicability of the theory by relaxing compactness and boundedness assumptions. In the context of financial mathematics, the relaxed assumptions are satisfied under the well-known no-arbitrage condition and the ``reasonable asymptotic elasticity'' condition of the utility function. Besides financial mathematics, we obtain several new results in linear and nonlinear stochastic programming and stochastic optimal control.Data-driven approximation of contextual chance-constrained stochastic programshttps://zbmath.org/1522.900542023-12-07T16:00:11.105023Z"Rahimian, Hamed"https://zbmath.org/authors/?q=ai:rahimian.hamed"Pagnoncelli, Bernardo"https://zbmath.org/authors/?q=ai:pagnoncelli.bernardo-kSummary: Uncertainty in classical stochastic programming models is often described solely by independent random parameters, ignoring their dependence on multidimensional features. We describe a novel contextual chance-constrained programming formulation that incorporates features, and argue that solutions that do not take them into account may not be implementable. Our formulation cannot be solved exactly in most cases, and we propose a tractable and fully data-driven approximate model that relies on weighted sums of random variables. We obtain a stochastic lower bound for the optimal value and feasibility results that include convergence to the true feasible set as the number of data points increases, as well as the minimal number of data points needed to obtain a feasible solution with high probability. We illustrate our findings in a vaccine allocation problem and compare the results with a naïve sample average approximation approach.Consistent approximations in composite optimizationhttps://zbmath.org/1522.900552023-12-07T16:00:11.105023Z"Royset, Johannes O."https://zbmath.org/authors/?q=ai:royset.johannes-oSummary: Approximations of optimization problems arise in computational procedures and sensitivity analysis. The resulting effect on solutions can be significant, with even small approximations of components of a problem translating into large errors in the solutions. We specify conditions under which approximations are well behaved in the sense of minimizers, stationary points, and level-sets and this leads to a framework of consistent approximations. The framework is developed for a broad class of composite problems, which are neither convex nor smooth. We demonstrate the framework using examples from stochastic optimization, neural-network based machine learning, distributionally robust optimization, penalty and augmented Lagrangian methods, interior-point methods, homotopy methods, smoothing methods, extended nonlinear programming, difference-of-convex programming, and multi-objective optimization. An enhanced proximal method illustrates the algorithmic possibilities. A quantitative analysis supplements the development by furnishing rates of convergence.Nonsmooth optimization by Lie bracket approximations into random directionshttps://zbmath.org/1522.900562023-12-07T16:00:11.105023Z"Suttner, Raik"https://zbmath.org/authors/?q=ai:suttner.raikSummary: We propose and analyze a novel extremum seeking control law for nonsmooth optimization. Our method is applicable to continuous-time input-affine systems on arbitrary state manifolds. The approach is based on a suitable combination of ideas from random direction stochastic approximation and Lie bracket approximation. Our main result provides sufficient conditions to ensure that the system state is driven towards a minimum of a locally Lipschitz continuous objective function. Under these conditions, we show that the closed-loop system approximates the behavior of a gradient-like system for a smoothed objective function. To prove the approximation property, we apply a stochastic averaging analysis with respect to the random control directions and a deterministic averaging analysis with respect to the employed periodic perturbation signals.Comparing stage-scenario with nodal formulation for multistage stochastic problemshttps://zbmath.org/1522.900572023-12-07T16:00:11.105023Z"Vitali, Sebastiano"https://zbmath.org/authors/?q=ai:vitali.sebastiano"Domínguez, Ruth"https://zbmath.org/authors/?q=ai:dominguez.ruth"Moriggia, Vittorio"https://zbmath.org/authors/?q=ai:moriggia.vittorioSummary: To solve real life problems under uncertainty in Economics, Finance, Energy, Transportation and Logistics, the use of stochastic optimization is widely accepted and appreciated. However, the nature of stochastic programming leads to a conflict between adaptability to reality and tractability. To formulate a multistage stochastic model, two types of formulations are typically adopted: the so-called \textit{stage-scenario formulation} named also \textit{formulation with explicit non-anticipativity constraints} and the so-called \textit{nodal formulation} named also \textit{formulation with implicit non-anticipativity constraints}. Both of them have advantages and disadvantages. This work aims at helping the scholars and practitioners to understand the two types of notation and, in particular, to reformulate with the nodal formulation a model that was originally defined with the stage-scenario formulation presenting this implementation in the algebraic language GAMS. In addition, this work presents an empirical analysis applying the two formulations both without any further decomposition to perform a fair comparison. In this way, we show that the difficulties to implement the model with the \textit{nodal formulation} are somehow reworded making the problem tractable without any decomposition algorithm. Still, we remark that in some other applications the \textit{stage-scenario formulation} could be more helpful to understand the structure of the problem since it allows to relax the non-anticipativity constraints.Solving two-stage stochastic variational inequalities by a hybrid projection semismooth Newton algorithmhttps://zbmath.org/1522.900582023-12-07T16:00:11.105023Z"Wang, Xiaozhou"https://zbmath.org/authors/?q=ai:wang.xiaozhou"Chen, Xiaojun"https://zbmath.org/authors/?q=ai:chen.xiaojun|chen.xiaojun.1Summary: A hybrid projection semismooth Newton algorithm (PSNA) is developed for solving two-stage stochastic variational inequalities; the algorithm is globally and superlinearly convergent under suitable assumptions. PSNA is a hybrid algorithm of the semismooth Newton algorithm and extragradient algorithm. At each step of PSNA, the second stage problem is split into a number of small variational inequality problems and solved in parallel for a fixed first stage decision iterate. The projection algorithm and semismooth Newton algorithm are used to find a new first stage decision iterate. Numerical results for large-scale nonmonotone two-stage stochastic variational inequalities and applications in traffic assignments show the efficiency of PSNA.Regularization for Wasserstein distributionally robust optimizationhttps://zbmath.org/1522.900592023-12-07T16:00:11.105023Z"Azizian, Waïss"https://zbmath.org/authors/?q=ai:azizian.waiss"Iutzeler, Franck"https://zbmath.org/authors/?q=ai:iutzeler.franck"Malick, Jérôme"https://zbmath.org/authors/?q=ai:malick.jeromeSummary: Optimal transport has recently proved to be a useful tool in various machine learning applications needing comparisons of probability measures. Among these, applications of distributionally robust optimization naturally involve Wasserstein distances in their models of uncertainty, capturing data shifts or worst-case scenarios. Inspired by the success of the regularization of Wasserstein distances in optimal transport, we study in this paper the regularization of Wasserstein distributionally robust optimization. First, we derive a general strong duality result of regularized Wasserstein distributionally robust problems. Second, we refine this duality result in the case of entropic regularization and provide an approximation result when the regularization parameters vanish.Robust optimization with continuous decision-dependent uncertainty with applications to demand response managementhttps://zbmath.org/1522.900602023-12-07T16:00:11.105023Z"Chen, Hongfan (Kevin)"https://zbmath.org/authors/?q=ai:chen.hongfan"Sun, Xu Andy"https://zbmath.org/authors/?q=ai:sun.xu-andy"Yang, Haoxiang"https://zbmath.org/authors/?q=ai:yang.haoxiangSummary: We consider a robust optimization problem with continuous decision-dependent uncertainty (RO-CDDU), which has two significant features: an uncertainty set linearly dependent on \textit{continuous} decision variables, and a convex piecewise-linear objective function. We prove that RO-CDDU is strongly \(\mathcal{NP}\)-hard in general and reformulate it into an equivalent mixed-integer nonlinear program (MINLP) with a decomposable structure to address the computational challenges. Such an MINLP model can be further transformed into a mixed-integer linear program (MILP) using extreme points of the dual polyhedron of the uncertainty set. We propose an alternating direction algorithm and a column generation algorithm for RO-CDDU. We model a robust demand response (DR) management problem in electricity markets as RO-CDDU, where electricity demand reduction from users is uncertain and depends on the DR planning decision. Extensive computational results demonstrate the promising performance of the proposed algorithms in both speed and solution quality. The results also shed light on how different magnitudes of decision-dependent uncertainty affect the DR decision.Approximating the chance-constrained capacitated vehicle routing problem with robust optimizationhttps://zbmath.org/1522.900612023-12-07T16:00:11.105023Z"Thiebaut, Karina"https://zbmath.org/authors/?q=ai:thiebaut.karina"Pessoa, Artur"https://zbmath.org/authors/?q=ai:pessoa.artur-alvesSummary: The Capacitated Vehicle Routing Problem (CVRP) is a classical combinatorial optimization problem that aims to serve a set of customers, using a set of identical vehicles, satisfying the vehicle capacities, and minimizing the total traveling distance. Among the possible approaches to extend the CVRP for handling uncertain demands, we highlight the robust optimization with budgeted uncertainty, and chance-constrained optimization. Another simpler and often omitted option is to apply the deterministic CVRP model over augmented demands in such a way to reduce the capacity violation probability. In this paper, we propose a suitable way to adjust the input data of both the deterministic CVRP and the robust CVRP with budgeted uncertainty so that the corresponding output approximates the chance-constrained CVRP for the case of independently normally distributed demands. In order to test our approach, we present quite extensive experiments showing that it leads to very small deviations with respect to the optimal chance-constrained solutions, and that the robust model brings significant benefits with respect to the deterministic one. In order to optimally solve the proposed chance-constrained benchmark instances, we also introduce a new technique to tighten a family of known inequalities for this problem.Correction to: ``Decision rule-based method in solving adjustable robust capacity expansion problem''https://zbmath.org/1522.900622023-12-07T16:00:11.105023Z"Zhao, Sixiang"https://zbmath.org/authors/?q=ai:zhao.sixiangA formula in the second paragraph of ``Problem description'' in the author's paper [ibid. 97, No. 2, 259--286 (2023; Zbl 1519.90145)] is corrected.Exact SDP relaxations for quadratic programs with bipartite graph structureshttps://zbmath.org/1522.900632023-12-07T16:00:11.105023Z"Azuma, Godai"https://zbmath.org/authors/?q=ai:azuma.godai"Fukuda, Mituhiro"https://zbmath.org/authors/?q=ai:fukuda.mituhiro"Kim, Sunyoung"https://zbmath.org/authors/?q=ai:kim.sunyoung"Yamashita, Makoto"https://zbmath.org/authors/?q=ai:yamashita.makoto.1Summary: For nonconvex quadratically constrained quadratic programs (QCQPs), we first show that, under certain feasibility conditions, the standard semidefinite programming (SDP) relaxation is exact for QCQPs with bipartite graph structures. The exact optimal solutions are obtained by examining the dual SDP relaxation and the rank of the optimal solution of this dual SDP relaxation under strong duality. Our results generalize the previous results on QCQPs with sign-definite bipartite graph structures, QCQPs with forest structures, and QCQPs with nonpositive off-diagonal data elements. Second, we propose a conversion method from QCQPs with no particular structure to the ones with bipartite graph structures. As a result, we demonstrate that a wider class of QCQPs can be exactly solved by the SDP relaxation. Numerical instances are presented for illustration.Optimal seating assignment in the COVID-19 era via quantum computinghttps://zbmath.org/1522.900642023-12-07T16:00:11.105023Z"Gioda, Ilaria"https://zbmath.org/authors/?q=ai:gioda.ilaria"Caputo, Davide"https://zbmath.org/authors/?q=ai:caputo.davide"Fadda, Edoardo"https://zbmath.org/authors/?q=ai:fadda.edoardo"Manerba, Daniele"https://zbmath.org/authors/?q=ai:manerba.daniele"Fernández, Blanca Silva"https://zbmath.org/authors/?q=ai:fernandez.blanca-silva"Tadei, Roberto"https://zbmath.org/authors/?q=ai:tadei.robertoSummary: In recent years, researchers have oriented their studies towards new technologies based on quantum physics that should allow the resolution of complex problems currently considered to be intractable. This new research area is called Quantum Computing. What makes Quantum Computing so attractive is the particular way with which quantum technology operates and the great potential it can offer to solve real-world problems. This work focuses on solving combinatorial optimization problems, specifically assignment problems, by exploiting this novel computational approach. A case-study, denoted as the Seating Arrangement Optimization problem, is considered. It is modeled through the Quadratic Unconstrained Binary Optimization (QUBO) paradigm and solved through two tools made available by the \textit{D-Wave Systems} company, QBSolv and a quantum-classical hybrid system. The obtained experimental results are compared in terms of solution quality and computational efficiency.
For the entire collection see [Zbl 1498.90005].A turnpike property for optimal control problems with dynamic probabilistic constraintshttps://zbmath.org/1522.900652023-12-07T16:00:11.105023Z"Gugat, Martin"https://zbmath.org/authors/?q=ai:gugat.martin"Heitsch, Holger"https://zbmath.org/authors/?q=ai:heitsch.holger"Henrion, René"https://zbmath.org/authors/?q=ai:henrion.reneSummary: We consider systems that are governed by linear time-discrete dynamics with an initial condition and a terminal condition for the expected values. We study optimal control problems where in the objective function a term of tracking type for the expected values and a control cost appear. In addition, the feasible states have to satisfy a conservative probabilistic constraint that requires that the probability that the trajectories remain in a given set \(F\) is greater than or equal to a given lower bound. An application are optimal control problems related to storage management systems with uncertain in- and output. We give sufficient conditions that imply that the optimal expected trajectories remain close to a certain state that can be characterized as the solution of an optimal control problem without prescribed initial- and terminal condition. In this way we contribute to the study of the turnpike phenomenon that is well-known in mathematical economics and make a step towards the extension of the turnpike theory to problems with probabilistic constraints.Accelerating inexact successive quadratic approximation for regularized optimization through manifold identificationhttps://zbmath.org/1522.900662023-12-07T16:00:11.105023Z"Lee, Ching-pei"https://zbmath.org/authors/?q=ai:lee.ching-peiSummary: For regularized optimization that minimizes the sum of a smooth term and a regularizer that promotes structured solutions, inexact proximal-Newton-type methods, or successive quadratic approximation (SQA) methods, are widely used for their superlinear convergence in terms of iterations. However, unlike the counter parts in smooth optimization, they suffer from lengthy running time in solving regularized subproblems because even approximate solutions cannot be computed easily, so their empirical time cost is not as impressive. In this work, we first show that for partly smooth regularizers, although general inexact solutions cannot identify the active manifold that makes the objective function smooth, approximate solutions generated by commonly-used subproblem solvers will identify this manifold, even with arbitrarily low solution precision. We then utilize this property to propose an improved SQA method, \(\mathrm{ISQA}^+\), that switches to efficient smooth optimization methods after this manifold is identified. We show that for a wide class of degenerate solutions, \(\mathrm{ISQA}^+\) possesses superlinear convergence not only in iterations, but also in running time because the cost per iteration is bounded. In particular, our superlinear convergence result holds on problems satisfying a sharpness condition that is more general than that in existing literature. We also prove iterate convergence under a sharpness condition for inexact SQA, which is novel for this family of methods that could easily violate the classical relative-error condition frequently used in proving convergence under similar conditions. Experiments on real-world problems support that \(\mathrm{ISQA}^+\) improves running time over some modern solvers for regularized optimization.Effective algorithms for separable nonconvex quadratic programming with one quadratic and box constraintshttps://zbmath.org/1522.900672023-12-07T16:00:11.105023Z"Luo, Hezhi"https://zbmath.org/authors/?q=ai:luo.hezhi"Zhang, Xianye"https://zbmath.org/authors/?q=ai:zhang.xianye"Wu, Huixian"https://zbmath.org/authors/?q=ai:wu.huixian"Xu, Weiqiang"https://zbmath.org/authors/?q=ai:xu.weiqiangSummary: We consider in this paper a separable and nonconvex quadratic program (QP) with a quadratic constraint and a box constraint that arises from application in optimal portfolio deleveraging (OPD) in finance and is known to be NP-hard. We first propose an improved Lagrangian breakpoint search algorithm based on the secant approach (called ILBSSA) for this nonconvex QP, and show that it converges to either a suboptimal solution or a global solution of the problem. We then develop a successive convex optimization (SCO) algorithm to improve the quality of suboptimal solutions derived from ILBSSA, and show that it converges to a KKT point of the problem. Second, we develop a new global algorithm (called ILBSSA-SCO-BB), which integrates the ILBSSA and SCO methods, convex relaxation and branch-and-bound framework, to find a globally optimal solution to the underlying QP within a pre-specified \(\epsilon\)-tolerance. We establish the convergence of the ILBSSA-SCO-BB algorithm and its complexity. Preliminary numerical results are reported to demonstrate the effectiveness of the ILBSSA-SCO-BB algorithm in finding a globally optimal solution to large-scale OPD instances.A Lagrangian-based approach for universum twin bounded support vector machine with its applicationshttps://zbmath.org/1522.900682023-12-07T16:00:11.105023Z"Moosaei, Hossein"https://zbmath.org/authors/?q=ai:moosaei.hossein"Hladík, Milan"https://zbmath.org/authors/?q=ai:hladik.milanSummary: The Universum provides prior knowledge about data in the mathematical problem to improve the generalization performance of the classifiers. Several works have shown that the Universum twin support vector machine \((\mathfrak{U}\)-TSVM) is an efficient method for binary classification problems. In this paper, we improve the \(\mathfrak{U}\)-TSVM method and propose an improved Universum twin bounded support vector machine (named as IUTBSVM). Indeed, by introducing different Lagrangian functions for the primal problems, we obtain new dual formulations of \(\mathfrak{U}\)-TSVM so that we do not need to compute inverse matrices. To reduce the computational time of the proposed method, we suggest a smaller size of the rectangular kernel matrices than the other methods. Numerical experiments on gender classification of human faces, handwritten digits recognition, and several UCI benchmark data sets indicate that the IUTBSVM is more efficient than the other four algorithms, namely \(\mathfrak{U}\)-SVM, TSVM, \(\mathfrak{U}\)-TSVM, and IUTSVM in the sense of the classification accuracy.Implicit regularity and linear convergence rates for the generalized trust-region subproblemhttps://zbmath.org/1522.900692023-12-07T16:00:11.105023Z"Wang, Alex L."https://zbmath.org/authors/?q=ai:wang.alex-l"Lu, Yunlei"https://zbmath.org/authors/?q=ai:lu.yunlei"Kilinç-Karzan, Fatma"https://zbmath.org/authors/?q=ai:kilinc-karzan.fatmaSummary: In this paper we develop efficient first-order algorithms for the generalized trust-region subproblem (GTRS), which has applications in signal processing, compressed sensing, and engineering. Although the GTRS, as stated, is nonlinear and nonconvex, it is well known that objective value exactness holds for its semidefinite programming (SDP) relaxation under a Slater condition. While polynomial-time SDP-based algorithms exist for the GTRS, their relatively large computational complexity has motivated and spurred the development of custom approaches for solving the GTRS. In particular, recent work in this direction has developed first-order methods for the GTRS whose running times are linear in the sparsity (the number of nonzero entries) of the input data. In contrast to these algorithms, in this paper we develop algorithms for computing \(\epsilon\)-approximate solutions to the GTRS whose running times are linear in both the input sparsity and the precision \(\log (1/\epsilon)\) whenever a regularity parameter is positive. We complement our theoretical guarantees with numerical experiments comparing our approach against algorithms from the literature. Our numerical experiments highlight that our new algorithms significantly outperform prior state-of-the-art algorithms on sparse large-scale instances.A sequential quadratic programming algorithm for nonsmooth problems with upper-\(\mathcal{C}^2\) objectivehttps://zbmath.org/1522.900702023-12-07T16:00:11.105023Z"Wang, Jingyi"https://zbmath.org/authors/?q=ai:wang.jingyi"Petra, Cosmin G."https://zbmath.org/authors/?q=ai:petra.cosmin-gSummary: An optimization algorithm for nonsmooth nonconvex constrained optimization problems with upper-\(\mathcal{C}^2\) objective functions is proposed and analyzed. Upper-\(\mathcal{C}^2\) is a weakly concave property that exists in difference of convex (DC) functions and arises naturally in many applications, particularly certain classes of solutions to parametric optimization problems e.g., recourse of stochastic programming and projection onto closed sets. The algorithm can be viewed as an extension of sequential quadratic programming (SQP) to nonsmooth problems with upper-\(\mathcal{C}^2\) objectives or a simplified bundle method. It is globally convergent with bounded algorithm parameters that are updated with a trust-region criterion. The algorithm handles general smooth constraints through linearization and uses a line search to ensure progress. The potential inconsistencies from the linearization of the constraints are addressed through a penalty method. The capabilities of the algorithm are demonstrated by solving both simple upper-\(\mathcal{C}^2\) problems and a real-world optimal power flow problem used in current power grid industry practices.A partial ellipsoidal approximation scheme for nonconvex homogeneous quadratic optimization with quadratic constraintshttps://zbmath.org/1522.900712023-12-07T16:00:11.105023Z"Xu, Zhuoyi"https://zbmath.org/authors/?q=ai:xu.zhuoyi"Li, Linbin"https://zbmath.org/authors/?q=ai:li.linbin"Xia, Yong"https://zbmath.org/authors/?q=ai:xia.yongSummary: An efficient partial ellipsoid approximation scheme is presented to find a \(\frac{1}{\lceil{\frac{m}{2}}\rceil } \)-approximation solution to the nonconvex homogeneous quadratic optimization with \(m\) convex quadratic constraints, where \(\lceil x \rceil\) is the smallest integer larger than or equal to \(x\). If there is an additional nonconvex quadratic constraint beyond the \(m\) convex constraints, we can use the new scheme to find a \(\frac{1}{m} \)-approximation solution.A stabilized sequential quadratic programming method for optimization problems in function spaceshttps://zbmath.org/1522.900722023-12-07T16:00:11.105023Z"Yamakawa, Yuya"https://zbmath.org/authors/?q=ai:yamakawa.yuyaSummary: In this paper, we propose a stabilized sequential quadratic programming (SQP) method for optimization problems in function spaces. A form of the problem considered in this paper can widely formulate many types of applications, such as obstacle problems, optimal control problems, and so on. Moreover, the proposed method is based on the existing stabilized SQP method and can find a point satisfying the Karush-Kuhn-Tucker (KKT) or asymptotic KKT conditions. One of the remarkable points is that we prove its global convergence to such a point under some assumptions without any constraint qualifications. In addition, we guarantee that an arbitrary accumulation point generated by the proposed method satisfies the KKT conditions under several additional assumptions. Finally, we report some numerical experiments to examine the effectiveness of the proposed method.Complexity analysis of interior-point methods for second-order stationary points of nonlinear semidefinite optimization problemshttps://zbmath.org/1522.900732023-12-07T16:00:11.105023Z"Arahata, Shun"https://zbmath.org/authors/?q=ai:arahata.shun"Okuno, Takayuki"https://zbmath.org/authors/?q=ai:okuno.takayuki"Takeda, Akiko"https://zbmath.org/authors/?q=ai:takeda.akikoSummary: We propose a primal-dual interior-point method (IPM) with convergence to second-order stationary points (SOSPs) of nonlinear semidefinite optimization problems, abbreviated as NSDPs. As far as we know, the current algorithms for NSDPs only ensure convergence to first-order stationary points such as Karush-Kuhn-Tucker points, but without a worst-case iteration complexity. The proposed method generates a sequence approximating SOSPs while minimizing a primal-dual merit function for NSDPs by using scaled gradient directions and directions of negative curvature. Under some assumptions, the generated sequence accumulates at an SOSP with a worst-case iteration complexity. This result is also obtained for a primal IPM with a slight modification. Finally, our numerical experiments show the benefits of using directions of negative curvature in the proposed method.Principled analyses and design of first-order methods with inexact proximal operatorshttps://zbmath.org/1522.900742023-12-07T16:00:11.105023Z"Barré, Mathieu"https://zbmath.org/authors/?q=ai:barre.mathieu"Taylor, Adrien B."https://zbmath.org/authors/?q=ai:taylor.adrien-b"Bach, Francis"https://zbmath.org/authors/?q=ai:bach.francis-rSummary: \textit{Proximal} operations are among the most common primitives appearing in both practical and theoretical (or high-level) optimization methods. This basic operation typically consists in solving an intermediary (hopefully simpler) optimization problem. In this work, we survey notions of inaccuracies that can be used when solving those intermediary optimization problems. Then, we show that worst-case guarantees for algorithms relying on such inexact proximal operations can be systematically obtained through a generic procedure based on semidefinite programming. This methodology is primarily based on the approach introduced by \textit{Y. Drori} and \textit{M. Teboulle} [Math. Program. 145, No. 1--2 (A), 451--482 (2014; Zbl 1300.90068)] and on convex interpolation results, and allows producing non-improvable worst-case analyses. In other words, for a given algorithm, the methodology generates both worst-case certificates (i.e., proofs) and problem instances on which they are achieved. Relying on this methodology, we study numerical worst-case performances of a few basic methods relying on inexact proximal operations including accelerated variants, and design a variant with optimized worst-case behavior. We further illustrate how to extend the approach to support strongly convex objectives by studying a simple relatively inexact proximal minimization method.A new perspective on low-rank optimizationhttps://zbmath.org/1522.900752023-12-07T16:00:11.105023Z"Bertsimas, Dimitris"https://zbmath.org/authors/?q=ai:bertsimas.dimitris-j"Cory-Wright, Ryan"https://zbmath.org/authors/?q=ai:cory-wright.ryan"Pauphilet, Jean"https://zbmath.org/authors/?q=ai:pauphilet.jeanSummary: A key question in many low-rank problems throughout optimization, machine learning, and statistics is to characterize the convex hulls of simple low-rank sets and judiciously apply these convex hulls to obtain strong yet computationally tractable relaxations. We invoke the matrix perspective function -- the matrix analog of the perspective function -- to characterize explicitly the convex hull of epigraphs of simple matrix convex functions under low-rank constraints. Further, we combine the matrix perspective function with orthogonal projection matrices -- the matrix analog of binary variables which capture the row-space of a matrix -- to develop a matrix perspective reformulation technique that reliably obtains strong relaxations for a variety of low-rank problems, including reduced rank regression, non-negative matrix factorization, and factor analysis. Moreover, we establish that these relaxations can be modeled via semidefinite constraints and thus optimized over tractably. The proposed approach parallels and generalizes the perspective reformulation technique in mixed-integer optimization and leads to new relaxations for a broad class of problems.DC semidefinite programming and cone constrained DC optimization. II: Local search methodshttps://zbmath.org/1522.900762023-12-07T16:00:11.105023Z"Dolgopolik, M. V."https://zbmath.org/authors/?q=ai:dolgopolik.maxim-vladimirovichSummary: The second part of our study is devoted to a detailed convergence analysis of two extensions of the well-known DCA method for solving DC (Difference of Convex functions) optimization problems to the case of general cone constrained DC optimization problems. We study the global convergence of the DCA for cone constrained problems and present a comprehensive analysis of a version of the DCA utilizing exact penalty functions. In particular, we study the exactness property of the penalized convex subproblems and provide two types of sufficient conditions for the convergence of the exact penalty method to a feasible and critical point of a cone constrained DC optimization problem from an infeasible starting point. In the numerical section of this work, the exact penalty DCA is applied to the problem of computing compressed modes for variational problems and the sphere packing problem on Grassmannian.
For Part I, see [Comput. Optim. Appl. 82, No. 3, 649--671 (2022; Zbl 1489.90106)].Empirical properties of optima in free semidefinite programshttps://zbmath.org/1522.900772023-12-07T16:00:11.105023Z"Evert, Eric"https://zbmath.org/authors/?q=ai:evert.eric"Fu, Yi"https://zbmath.org/authors/?q=ai:fu.yi"Helton, J. William"https://zbmath.org/authors/?q=ai:helton.john-william"Yin, John"https://zbmath.org/authors/?q=ai:yin.john-bohan|yin.johnSummary: Semidefinite programming is based on optimization of linear functionals over convex sets defined by linear matrix inequalities, namely, inequalities of the form
\[
L_A(X)=I-A_1X_1-\cdots -A_gX_g\succeq 0.
\]
Here, the \(X_j\) are real numbers and the set of solutions is called a spectrahedron. These inequalities make sense when the \(X_i\) are symmetric matrices of any size, \(n \times n\), and enter the formula though tensor product \(A_i\otimes X_i\): The solution set of \(L_A(X)\succeq 0\) is called a free spectrahedron since it contains matrices of all sizes and the defining ``linear pencil'' is ``free'' of the sizes of the matrices. In this article, we report on empirically observed properties of optimizers obtained from optimizing linear functionals over free spectrahedra restricted to matrices \(X_i\) of fixed size \(n \times n\). The optimizers we find are always classical extreme points. Surprisingly, in many reasonable parameter ranges, over 99.9\% are also free extreme points. Moreover, the dimension of the active constraint, \(\ker(L_A(X^\ell))\), is about twice what we expected. Another distinctive pattern regards reducibility of optimizing tuples \( (X^\ell_1,\dots,X^\ell_g)\). We give an algorithm for representing elements of a free spectrahedron as matrix convex combinations of free extreme points; these representations satisfy a very low bound on the number of free extreme points needed.A spectral method for joint community detection and orthogonal group synchronizationhttps://zbmath.org/1522.900782023-12-07T16:00:11.105023Z"Fan, Yifeng"https://zbmath.org/authors/?q=ai:fan.yifeng"Khoo, Yuehaw"https://zbmath.org/authors/?q=ai:khoo.yuehaw"Zhao, Zhizhen"https://zbmath.org/authors/?q=ai:zhao.zhizhenSummary: Community detection and orthogonal group synchronization are both fundamental problems with a variety of important applications in science and engineering. In this work, we consider the joint problem of community detection and orthogonal group synchronization which aims to recover the communities and perform synchronization simultaneously. To this end, we propose a simple algorithm that consists of a spectral decomposition step followed by a blockwise column pivoted QR factorization. The proposed algorithm is efficient and scales linearly with the number of edges in the graph. We also leverage the recently developed ``leave-one-out'' technique to establish a near-optimal guarantee for exact recovery of the cluster memberships and stable recovery of the orthogonal transforms. Numerical experiments demonstrate the efficiency and efficacy of our algorithm and confirm our theoretical characterization of it.Inexact penalty decomposition methods for optimization problems with geometric constraintshttps://zbmath.org/1522.900792023-12-07T16:00:11.105023Z"Kanzow, Christian"https://zbmath.org/authors/?q=ai:kanzow.christian"Lapucci, Matteo"https://zbmath.org/authors/?q=ai:lapucci.matteoSummary: This paper provides a theoretical and numerical investigation of a penalty decomposition scheme for the solution of optimization problems with geometric constraints. In particular, we consider some situations where parts of the constraints are nonconvex and complicated, like cardinality constraints, disjunctive programs, or matrix problems involving rank constraints. By a variable duplication and decomposition strategy, the method presented here explicitly handles these difficult constraints, thus generating iterates which are feasible with respect to them, while the remaining (standard and supposingly simple) constraints are tackled by sequential penalization. Inexact optimization steps are proven sufficient for the resulting algorithm to work, so that it is employable even with difficult objective functions. The current work is therefore a significant generalization of existing papers on penalty decomposition methods. On the other hand, it is related to some recent publications which use an augmented Lagrangian idea to solve optimization problems with geometric constraints. Compared to these methods, the decomposition idea is shown to be numerically superior since it allows much more freedom in the choice of the subproblem solver, and since the number of certain (possibly expensive) projection steps is significantly less. Extensive numerical results on several highly complicated classes of optimization problems in vector and matrix spaces indicate that the current method is indeed very efficient to solve these problems.A unified approach to synchronization problems over subgroups of the orthogonal grouphttps://zbmath.org/1522.900802023-12-07T16:00:11.105023Z"Liu, Huikang"https://zbmath.org/authors/?q=ai:liu.huikang"Yue, Man-Chung"https://zbmath.org/authors/?q=ai:yue.man-chung"So, Anthony Man-Cho"https://zbmath.org/authors/?q=ai:so.anthony-man-choSummary: The problem of synchronization over a group \(\mathcal{G}\) aims to estimate a collection of group elements \(G_1^{\ast}, \ldots, G_n^{\ast} \in \mathcal{G}\) based on noisy observations of a subset of all pairwise ratios of the form \(G_i^{\ast} G_j^{\ast^{-1}}\). Such a problem has gained much attention recently and finds many applications across a wide range of scientific and engineering areas. In this paper, we consider the class of synchronization problems in which the group is a closed subgroup of the orthogonal group. This class covers many group synchronization problems that arise in practice. Our contribution is fivefold. First, we propose a unified approach for solving this class of group synchronization problems, which consists of a suitable initialization step and an iterative refinement step based on the generalized power method, and show that it enjoys a strong theoretical guarantee on the estimation error under certain assumptions on the group, measurement graph, noise, and initialization. Second, we formulate two geometric conditions that are required by our approach and show that they hold for various practically relevant subgroups of the orthogonal group. The conditions are closely related to the error-bound geometry of the subgroup -- an important notion in optimization. Third, we verify the assumptions on the measurement graph and noise for standard random graph and random matrix models. Fourth, based on the classic notion of metric entropy, we develop and analyze a novel spectral-type estimator. Finally, we show via extensive numerical experiments that our proposed non-convex approach outperforms existing approaches in terms of computational speed, scalability, and/or estimation error.The variant of primal simplex-type method for linear second-order cone programminghttps://zbmath.org/1522.900812023-12-07T16:00:11.105023Z"Zhadan, Vitaly"https://zbmath.org/authors/?q=ai:zhadan.vitalii-grigorevichSummary: The linear second-order cone programming problem is considered. For its solution the variant of the primal simplex-type method is proposed. The notion of the \textbf{S}-extreme point of the feasible set is introduced. Among all \textbf{S}-extreme points the regular \textbf{S}-extreme points are regarded separately. The passage from the regular \textbf{S}-extreme point to another one is described. The method can be treated as the generalization of the primal simplex-type method for linear programming. At each iteration the dual variable together with the dual weak variable are defined. As in linear programming, the basis of the extreme point is used. Among all basic variables the facet basic variables and the interior basic variable are selected.
For the entire collection see [Zbl 1508.90001].Low-rank univariate sum of squares has no spurious local minimahttps://zbmath.org/1522.900822023-12-07T16:00:11.105023Z"Legat, Benoît"https://zbmath.org/authors/?q=ai:legat.benoit"Yuan, Chenyang"https://zbmath.org/authors/?q=ai:yuan.chenyang"Parrilo, Pablo"https://zbmath.org/authors/?q=ai:parrilo.pablo-aSummary: We study the problem of decomposing a polynomial \(p\) into a sum of \(r\) squares by minimizing a quadratically penalized objective \(f_p(\mathbf{u})=\Vert\sum_{i=1}^ru_i^2-p\Vert^2\). This objective is nonconvex and is equivalent to the rank-\(r\) Burer-Monteiro factorization of a semidefinite program (SDP) encoding the sum of squares decomposition. We show that for all univariate polynomials \(p\), if \(r\geq 2\), then \(f_p(\mathbf{u})\) has no spurious second-order critical points, showing that all local optima are also global optima. This is in contrast to previous work showing that for general SDPs, in addition to genericity conditions, \(r\) has to be roughly the square root of the number of constraints (the degree of \(p)\) for there to be no spurious second-order critical points. Our proof uses tools from computational algebraic geometry and can be interpreted as constructing a certificate using the first- and second-order necessary conditions. We also show that by choosing a norm based on sampling equally spaced points on the circle, the gradient \(\nabla f_p\) can be computed in nearly linear time using fast Fourier transforms. Experimentally we demonstrate that this method has very fast convergence using first-order optimization algorithms such as L-BFGS, with near-linear scaling to million-degree polynomials.Rational generalized Nash equilibrium problemshttps://zbmath.org/1522.900832023-12-07T16:00:11.105023Z"Nie, Jiawang"https://zbmath.org/authors/?q=ai:nie.jiawang"Tang, Xindong"https://zbmath.org/authors/?q=ai:tang.xindong"Zhong, Suhan"https://zbmath.org/authors/?q=ai:zhong.suhanSummary: This paper studies generalized Nash equilibrium problems that are given by rational functions. The optimization problems are not assumed to be convex. Rational expressions for Lagrange multipliers and feasible extensions of KKT points are introduced to compute a generalized Nash equilibrium (GNE). We give a hierarchy of rational optimization problems to solve rational generalized Nash equilibrium problems. The existence and computation of feasible extensions are studied. The Moment-SOS relaxations are applied to solve the rational optimization problems. Under some general assumptions, we show that the proposed hierarchy can compute a GNE if it exists or detect its nonexistence. Numerical experiments are given to show the efficiency of the proposed method.Tropical complementarity problems and Nash equilibriahttps://zbmath.org/1522.900842023-12-07T16:00:11.105023Z"Allamigeon, Xavier"https://zbmath.org/authors/?q=ai:allamigeon.xavier"Gaubert, Stéphane"https://zbmath.org/authors/?q=ai:gaubert.stephane"Meunier, Frédéric"https://zbmath.org/authors/?q=ai:meunier.fredericSummary: Linear complementarity programming is a generalization of linear programming which encompasses the computation of Nash equilibria for bimatrix games. While the latter problem is PPAD-complete, we show that the tropical analogue of the complementarity problem associated with Nash equilibria can be solved in polynomial time. Moreover, we prove that the Lemke-Howson algorithm carries over the tropical setting and performs a linear number of pivots in the worst case. A consequence of this result is a new class of (classical) bimatrix games for which Nash equilibria computation can be done in polynomial time.A Newton-type proximal gradient method for nonlinear multi-objective optimization problemshttps://zbmath.org/1522.900852023-12-07T16:00:11.105023Z"Ansary, Md Abu Talhamainuddin"https://zbmath.org/authors/?q=ai:ansary.md-abu-talhamainuddinSummary: In this paper, a globally convergent Newton-type proximal gradient method is developed for composite multi-objective optimization problems where each objective function can be represented as the sum of a smooth function and a nonsmooth function. The proposed method deals with unconstrained convex multi-objective optimization problems. This method is free from any kind of priori chosen parameters or ordering information of objective functions. At every iteration of the proposed method, a subproblem is solved to find a suitable descent direction. The subproblem uses a quadratic approximation of each smooth function. An Armijo type line search is conducted to find a suitable step length. A sequence is generated using the descent direction and the step length. The global convergence of this method is justified under some mild assumptions. The proposed method is verified and compared with some existing methods using a set of test problems.The homogenization cone: polar cone and projectionhttps://zbmath.org/1522.900862023-12-07T16:00:11.105023Z"Bauschke, Heinz H."https://zbmath.org/authors/?q=ai:bauschke.heinz-h"Bendit, Theo"https://zbmath.org/authors/?q=ai:bendit.theo"Wang, Hansen"https://zbmath.org/authors/?q=ai:wang.hansenSummary: Let \(C\) be a closed convex subset of a real Hilbert space containing the origin, and assume that \(K\) is the homogenization cone of \(C\), i.e., the smallest closed convex cone containing \(C\times \{1\}\). Homogenization cones play an important role in optimization for the construction of examples and counterexamples. A famous examples is the second-order/Lorentz/``ice cream'' cone which is the homogenization cone of the unit ball. In this paper, we discuss the polar cone of \(K\) as well as an algorithm for finding the projection onto \(K\) provided that the projection onto \(C\) is available. Various examples illustrate our results.Time-varying distributed optimization problem with inequality constraintshttps://zbmath.org/1522.900872023-12-07T16:00:11.105023Z"Chen, Yong"https://zbmath.org/authors/?q=ai:chen.yong.4"Yu, Tao"https://zbmath.org/authors/?q=ai:yu.tao.2"Meng, Qing"https://zbmath.org/authors/?q=ai:meng.qing|meng.qing.1"Niu, Fuxi"https://zbmath.org/authors/?q=ai:niu.fuxi"Wang, Haibo"https://zbmath.org/authors/?q=ai:wang.haiboSummary: This paper discusses a distributed time-varying convex optimization problem that agents have different Hessian matrices with inequality constraints. The objective is to minimize the sum of local time-varying objective functions of agents constrained by time-varying inequalities. Under the condition of undirected connected graph, a distributed continuous time consistency algorithm is designed based on average consensus estimator, sign function and log-barrier penalty function. The main idea of the algorithm proposed in this paper is to use the estimator to estimate the global information, make the state of agents reach consensus and achieve gradient descent to track the optimal solution. Theoretical findings show that all agents can reach an agreement, the proposed algorithm can track the optimal solution of the time-varying optimization problem. The effectiveness of the theoretical results is verified through a numerical examples.Random coordinate descent methods for nonseparable composite optimizationhttps://zbmath.org/1522.900882023-12-07T16:00:11.105023Z"Chorobura, Flavia"https://zbmath.org/authors/?q=ai:chorobura.flavia"Necoara, Ion"https://zbmath.org/authors/?q=ai:necoara.ionSummary: In this paper we consider large-scale composite optimization problems having the objective function formed as a sum of two terms (possibly nonconvex); one has a (block) coordinatewise Lipschitz continuous gradient and the other is differentiable but nonseparable. Under these general settings we derive and analyze two new coordinate descent methods. The first algorithm, referred to as the coordinate proximal gradient method, considers the composite form of the objective function, while the other algorithm disregards the composite form of the objective and uses the partial gradient of the full objective, yielding a coordinate gradient descent scheme with novel adaptive stepsize rules. We prove that these new stepsize rules make the coordinate gradient scheme a descent method, provided that additional assumptions hold for the second term in the objective function. We present a complete worst-case complexity analysis for these two new methods in both convex and nonconvex settings, provided that the (block) coordinates are chosen random or cyclic. Preliminary numerical results also confirm the efficiency of our two algorithms for practical problems.Distributed online bandit linear regressions with differential privacyhttps://zbmath.org/1522.900892023-12-07T16:00:11.105023Z"Dai, Mingcheng"https://zbmath.org/authors/?q=ai:dai.mingcheng"Ho, Daniel W. C."https://zbmath.org/authors/?q=ai:ho.daniel-w-c"Zhang, Baoyong"https://zbmath.org/authors/?q=ai:zhang.baoyong"Yuan, Deming"https://zbmath.org/authors/?q=ai:yuan.deming"Xu, Shengyuan"https://zbmath.org/authors/?q=ai:xu.shengyuanSummary: This paper addresses the distributed online bandit linear regression problems with privacy protection, in which the training data are spread in a multi-agent network. Each node identifies a linear predictor to fit the training data and experiences a square loss on each round. The purpose is to minimize the regret that assesses the difference of the accumulated loss between the online linear predictor and the optimal offline linear predictor. Moreover, the differential privacy strategy is adopted to prevent the adversary from inferring the parameter vector of any node. Two efficient differentially private distributed online regression algorithms are developed in the cases of one-point and two-point bandit feedback. Our analysis suggests that the developed algorithms achieve \(\epsilon\)-differential privacy and establish the regret upper bounds in \(\mathcal{O}(K^{3/4})\) and \(\mathcal{O}(\sqrt{K})\) for one-point and two-point bandit feedback, respectively, where \(K\) is the time horizon. We also show that there exists a tradeoff between our algorithms' privacy level and convergence. Finally, the performance of the proposed algorithms is validated by a numerical example.An accelerated proximal alternating direction method of multipliers for robust fused Lassohttps://zbmath.org/1522.900902023-12-07T16:00:11.105023Z"Fan, Yibao"https://zbmath.org/authors/?q=ai:fan.yibao"Shang, Youlin"https://zbmath.org/authors/?q=ai:shang.youlin"Jin, Zheng-Fen"https://zbmath.org/authors/?q=ai:jin.zhengfen"Liu, Jia"https://zbmath.org/authors/?q=ai:liu.jia.3|liu.jia.2"Zhang, Roxin"https://zbmath.org/authors/?q=ai:zhang.roxinSummary: In the era of big data, much of the data is susceptible to noise with heavy-tailed distribution. Fused Lasso can effectively handle high dimensional sparse data with strong correlation between two adjacent variables under known Gaussian noise. However, it has poor robustness to non-Gaussian noise with heavy-tailed distribution. Robust fused Lasso with \(l_1\) norm loss function can overcome the drawback of fused Lasso when noise is heavy-tailed distribution. But the key challenge for solving this model is nonsmoothness and its nonseparability. Therefore, in this paper, we first deform the robust fused Lasso into an easily solvable form, which changes the three-block objective function to a two-block form. Then, we propose an accelerated proximal alternating direction method of multipliers (APADMM) with an additional update step, which is base on a new PADMM that changes the Lagrangian multiplier term update. Furthermore, we give the \(O(1/K)\) nonergodic convergence rate analysis of the proposed APADMM. Finally, numerical results show that the proposed new PADMM and APADMM have better performance than other existing ADMM solvers.Spectral graph matching and regularized quadratic relaxations. I: Algorithm and Gaussian analysishttps://zbmath.org/1522.900912023-12-07T16:00:11.105023Z"Fan, Zhou"https://zbmath.org/authors/?q=ai:fan.zhou"Mao, Cheng"https://zbmath.org/authors/?q=ai:mao.cheng"Wu, Yihong"https://zbmath.org/authors/?q=ai:wu.yihong"Xu, Jiaming"https://zbmath.org/authors/?q=ai:xu.jiamingSummary: Graph matching aims at finding the vertex correspondence between two unlabeled graphs that maximizes the total edge weight correlation. This amounts to solving a computationally intractable quadratic assignment problem. In this paper, we propose a new spectral method, graph matching by pairwise eigen-alignments (GRAMPA). Departing from prior spectral approaches that only compare top eigenvectors, or eigenvectors of the same order, GRAMPA first constructs a similarity matrix as a weighted sum of outer products between \textit{all} pairs of eigenvectors of the two graphs, with weights given by a Cauchy kernel applied to the separation of the corresponding eigenvalues, then outputs a matching by a simple rounding procedure. The similarity matrix can also be interpreted as the solution to a regularized quadratic programming relaxation of the quadratic assignment problem. For the Gaussian Wigner model in which two complete graphs on \(n\) vertices have Gaussian edge weights with correlation coefficient \(1-\sigma^2\), we show that GRAMPA exactly recovers the correct vertex correspondence with high probability when \(\sigma = O(\frac{1}{\log n})\). This matches the state of the art of polynomial-time algorithms and significantly improves over existing spectral methods which require \(\sigma\) to be polynomially small in \(n\). The superiority of GRAMPA is also demonstrated on a variety of synthetic and real datasets, in terms of both statistical accuracy and computational efficiency. Universality results, including similar guarantees for dense and sparse Erdős-Rényi graphs, are deferred to a companion paper.
For Part II see [Found. Comput. Math. 23, No. 5, 1567--1617 (2023; Zbl 1522.90092)].Spectral graph matching and regularized quadratic relaxations. II: Erdős-Rényi graphs and universalityhttps://zbmath.org/1522.900922023-12-07T16:00:11.105023Z"Fan, Zhou"https://zbmath.org/authors/?q=ai:fan.zhou"Mao, Cheng"https://zbmath.org/authors/?q=ai:mao.cheng"Wu, Yihong"https://zbmath.org/authors/?q=ai:wu.yihong"Xu, Jiaming"https://zbmath.org/authors/?q=ai:xu.jiamingSummary: We analyze a new spectral graph matching algorithm, GRAph Matching by Pairwise eigen-Alignments (GRAMPA), for recovering the latent vertex correspondence between two unlabeled, edge-correlated weighted graphs. Extending the exact recovery guarantees established in a companion paper for Gaussian weights, in this work, we prove the universality of these guarantees for a general correlated Wigner model. In particular, for two Erdős-Rényi graphs with edge correlation coefficient \(1-\sigma^2\) and average degree at least \({\text{polylog}}(n)\), we show that GRAMPA exactly recovers the latent vertex correspondence with high probability when \(\sigma \lesssim 1/{\text{polylog}}(n)\). Moreover, we establish a similar guarantee for a variant of GRAMPA, corresponding to a tighter quadratic programming relaxation of the quadratic assignment problem. Our analysis exploits a resolvent representation of the GRAMPA similarity matrix and local laws for the resolvents of sparse Wigner matrices.
For Part I see [Found. Comput. Math. 23, No. 5, 1511--1565 (2023; Zbl 1522.90091)].Adaptive third-order methods for composite convex optimizationhttps://zbmath.org/1522.900932023-12-07T16:00:11.105023Z"Grapiglia, G. N."https://zbmath.org/authors/?q=ai:grapiglia.geovani-nunes"Nesterov, Yu."https://zbmath.org/authors/?q=ai:nesterov.yuriiSummary: In this paper we propose third-order methods for composite convex optimization problems in which the smooth part is a three-times continuously differentiable function with Lipschitz continuous third-order derivatives. The methods are adaptive in the sense that they do not require knowledge of the Lipschitz constant. Trial points are computed by the inexact minimization of models that consist in the nonsmooth part of the objective plus a quartic regularization of third-order Taylor polynomial of the smooth part. Specifically, approximate solutions of the auxiliary problems are obtained by using a Bregman gradient method as inner solver. Different from existing adaptive approaches for high-order methods, in our new schemes the regularization parameters are tuned by checking the progress of the inner solver. With this technique, we show that the basic method finds an \(\epsilon\)-approximate minimizer of the objective function performing at most \(\mathcal{O}(|\log(\epsilon)|\epsilon^{-\frac{1}{3}})\) iterations of the inner solver. An accelerated adaptive third-order method is also presented with total inner iteration complexity of \(\mathcal{O} (|\log(\epsilon)|\epsilon^{-\frac{1}{4}})\).Quasi-error bounds for \(p\)-convex set-valued mappingshttps://zbmath.org/1522.900942023-12-07T16:00:11.105023Z"Huang, Hui"https://zbmath.org/authors/?q=ai:huang.hui"Zhu, Jiangxing"https://zbmath.org/authors/?q=ai:zhu.jiangxingSummary: We first introduce the concept of \(p\)-convex set-valued mappings, which is an extension of \(p\)-convex functions. Then, we show that for a \(p\)-convex set optimization problem, a local Pareto minimizer is also a global Pareto minimizer. Moreover, we obtain some results of the type of Robinson-Ursescu theorem for \(p\)-convex set-valued mappings in Banach spaces. By adopting a new concept of quasi-error bound for set-valued mappings, we establish some results on the existence of quasi-error bounds for \(p\)-convex set-valued mappings.Adaptive catalyst for smooth convex optimizationhttps://zbmath.org/1522.900952023-12-07T16:00:11.105023Z"Ivanova, Anastasiya"https://zbmath.org/authors/?q=ai:ivanova.anastasiya"Pasechnyuk, Dmitry"https://zbmath.org/authors/?q=ai:pasechnyuk.dmitry-a"Grishchenko, Dmitry"https://zbmath.org/authors/?q=ai:grishchenko.dmitry"Shulgin, Egor"https://zbmath.org/authors/?q=ai:shulgin.egor"Gasnikov, Alexander"https://zbmath.org/authors/?q=ai:gasnikov.aleksandr-v"Matyukhin, Vladislav"https://zbmath.org/authors/?q=ai:matyukhin.vladislav-vSummary: In this paper, we present a generic framework that allows accelerating almost arbitrary non-accelerated deterministic and randomized algorithms for smooth convex optimization problems. The major approach of our \textit{envelope} is the same as in \textit{Catalyst} [37]: an accelerated proximal outer gradient method, which is used as an envelope for a non-accelerated inner method for the \(\ell_2\) regularized auxiliary problem. Our algorithm has two key differences: 1) easily verifiable stopping condition for inner algorithm; 2) the regularization parameter can be tuned along the way. As a result, the main contribution of our work is a new framework that applies to adaptive inner algorithms: Steepest Descent, Adaptive Coordinate Descent, Alternating Minimization. Moreover, in the non-adaptive case, our approach allows obtaining Catalyst without a logarithmic factor, which appears in the standard Catalyst [37, 38].
For the entire collection see [Zbl 1508.90001].A second-order accelerated neurodynamic approach for distributed convex optimizationhttps://zbmath.org/1522.900962023-12-07T16:00:11.105023Z"Jiang, Xinrui"https://zbmath.org/authors/?q=ai:jiang.xinrui"Qin, Sitian"https://zbmath.org/authors/?q=ai:qin.sitian"Xue, Xiaoping"https://zbmath.org/authors/?q=ai:xue.xiaoping"Liu, Xinzhi"https://zbmath.org/authors/?q=ai:liu.xinzhiSummary: Based on the theories of inertial systems, a second-order accelerated neurodynamic approach is designed to solve a distributed convex optimization with inequality and set constraints. Most of the existing approaches for distributed convex optimization problems are usually first-order ones, and it is usually hard to analyze the convergence rate for the state solution of those first-order approaches. Due to the control design for the acceleration, the second-order neurodynamic approaches can often achieve faster convergence rate. Moreover, the existing second-order approaches are mostly designed to solve unconstrained distributed convex optimization problems, and are not suitable for solving constrained distributed convex optimization problems. It is acquired that the state solution of the designed neurodynamic approach in this paper converges to the optimal solution of the considered distributed convex optimization problem. An error function which demonstrates the performance of the designed neurodynamic approach, has a superquadratic convergence. Several numerical examples are provided to show the effectiveness of the presented second-order accelerated neurodynamic approach.On weakly efficient solutions for semidefinite linear fractional vector optimization problemshttps://zbmath.org/1522.900972023-12-07T16:00:11.105023Z"Kim, Moon Hee"https://zbmath.org/authors/?q=ai:kim.moon-hee"Kim, Gwi Soo"https://zbmath.org/authors/?q=ai:kim.gwi-soo"Lee, Gue Myung"https://zbmath.org/authors/?q=ai:lee.gue-myungSummary: We consider a semidefinite linear fractional vector optimization problem (FVP) and establish optimality theorems for weakly efficient solutions for (FVP), which hold without any constraint qualification. We first discuss the relation between weakly efficient solution of (FVP) and one of its related linear vector optimization problem (LVP). By using the relation and the maximum function of objective functions of (FVP), we obtain our optimality theorems for weakly efficient solutions for (FVP), and then we give examples showing how to use our optimality theorems for finding weakly efficient solutions for (FVP). Moreover, we formulate vector dual problem (VD) for (FVP), which is a kind of vector version of Wolfe dual problem, and establish duality theorems for (FVP) and (VD), which hold without any constraint qualification.Sequential Pareto subdifferential sum rule for convex set-valued mappings and applicationshttps://zbmath.org/1522.900982023-12-07T16:00:11.105023Z"Laghdir, Mohamed"https://zbmath.org/authors/?q=ai:laghdir.mohamed"Echchaabaoui, El Mahjoub"https://zbmath.org/authors/?q=ai:echchaabaoui.el-mahjoubSummary: The aim of this paper is to provide a general description of the Pareto subdifferential (weak and proper) of the sum of two cone-convex set-valued mappings in terms of sequences without any constraint qualifications. As an application, we derive sequential Lagrange multipliers optimality conditions for general set-valued optimization problem in terms of sequential Lagrange multipliers at nearby points for the Pareto efficient solutions, where no constraint qualification is assumed.Graph topology invariant gradient and sampling complexity for decentralized and stochastic optimizationhttps://zbmath.org/1522.900992023-12-07T16:00:11.105023Z"Lan, Guanghui"https://zbmath.org/authors/?q=ai:lan.guanghui"Ouyang, Yuyuan"https://zbmath.org/authors/?q=ai:ouyang.yuyuan"Zhou, Yi"https://zbmath.org/authors/?q=ai:zhou.yi.6Summary: One fundamental problem in constrained decentralized multiagent optimization is the trade-off between gradient/sampling complexity and communication complexity. In this paper, we propose new algorithms whose gradient and sampling complexities are graph topology invariant, while their communication complexities remain optimal. Specifically, for convex smooth deterministic problems, we propose a primal-dual sliding (PDS) algorithm that is able to compute an \(\varepsilon\)-solution with \(\mathcal{O}((\tilde{L}/\varepsilon)^{1/2})\) gradient complexity and \(\mathcal{O}((\tilde{L}/\varepsilon)^{1/2}+\|\mathcal{A}\|/\varepsilon)\) communication complexity, where \(\tilde{L}\) is the smoothness parameter of the objective function and \(\mathcal{A}\) is related to either the graph Laplacian or the transpose of the oriented incidence matrix of the communication network. The complexities can be further improved to \(\mathcal{O}((\tilde{L}/\mu)^{1/2}\log (1/\varepsilon))\) and \(\mathcal{O}((\tilde{L}/\mu)^{1/2}\log(1/\varepsilon)+ \|\mathcal{A}\|/\varepsilon^{1/2})\), respectively, with the additional assumption of strong convexity modulus \(\mu\). We also propose a stochastic variant, namely, the stochastic primal-dual sliding (SPDS) algorithm, for convex smooth problems with stochastic gradients. The SPDS algorithm utilizes the minibatch technique and enables the agents to perform sampling and communication simultaneously. It computes a stochastic \(\varepsilon \)-solution with \(\mathcal{O}((\tilde{L}/\varepsilon)^{1/2}+ (\sigma/\varepsilon)^2)\) sampling complexity, which can be further improved to \(\mathcal{O}((\tilde{L}/\mu)^{1/2}\log(1/\varepsilon)+ \sigma^2/\varepsilon)\) in the strong convexity case. Here \(\sigma^2\) is the variance of the stochastic gradient. The communication complexities of SPDS remain the same as that of the deterministic case.Optimal methods for convex risk-averse distributed optimizationhttps://zbmath.org/1522.901002023-12-07T16:00:11.105023Z"Lan, Guanghui"https://zbmath.org/authors/?q=ai:lan.guanghui"Zhang, Zhe"https://zbmath.org/authors/?q=ai:zhang.zhe.2Summary: This paper studies the communication complexity of convex risk-averse optimization over a network. The problem generalizes the well-studied risk-neutral finite-sum distributed optimization problem, and its importance stems from the need to handle risk in an uncertain environment. For algorithms in the literature, a gap exists in communication complexities for solving risk-averse and risk-neutral problems. We propose two distributed algorithms, namely the distributed risk-averse optimization (DRAO) method and the distributed risk-averse optimization with sliding (DRAO-S) method, to close the gap. Specifically, the DRAO method achieves optimal communication complexity by assuming a certain saddle point subproblem can be easily solved in the server node. The DRAO-S method removes the strong assumption by introducing a novel saddle point sliding subroutine which only requires the projection over the ambiguity set \(P\). We observe that the number of \(P\)-projections performed by DRAO-S is optimal. Moreover, we develop matching lower complexity bounds to show the communication complexities of both DRAO and DRAO-S to be unimprovable. Numerical experiments are conducted to demonstrate the encouraging empirical performance of the DRAO-S method.Accelerated first-order methods for convex optimization with locally Lipschitz continuous gradienthttps://zbmath.org/1522.901012023-12-07T16:00:11.105023Z"Lu, Zhaosong"https://zbmath.org/authors/?q=ai:lu.zhaosong"Mei, Sanyou"https://zbmath.org/authors/?q=ai:mei.sanyouSummary: In this paper we develop accelerated first-order methods for convex optimization with \textit{locally Lipschitz} continuous gradient (LLCG), which is beyond the well-studied class of convex optimization with Lipschitz continuous gradient. In particular, we first consider unconstrained convex optimization with LLCG and propose accelerated proximal gradient (APG) methods for solving it. The proposed APG methods are equipped with a verifiable termination criterion and enjoy an operation complexity of \(\mathcal{O}(\varepsilon^{-1/2}\log\varepsilon^{-1})\) and \(\mathcal{O}(\log\varepsilon^{-1})\) for finding an \(\varepsilon\)-residual solution of an unconstrained convex and strongly convex optimization problem, respectively. We then consider constrained convex optimization with LLCG and propose a first-order proximal augmented Lagrangian method for solving it by applying one of our proposed APG methods to approximately solve a sequence of proximal augmented Lagrangian subproblems. The resulting method is equipped with a verifiable termination criterion and enjoys an operation complexity of \(\mathcal{O}(\varepsilon^{-1}\log \varepsilon^{-1})\) and \(\mathcal{O}(\varepsilon^{-1/2}\log\varepsilon^{-1})\) for finding an \(\varepsilon\)-KKT solution of a constrained convex and strongly convex optimization problem, respectively. All the proposed methods in this paper are \textit{parameter-free} or \textit{almost parameter-free} except that knowledge of the convexity parameter is required. In addition, preliminary numerical results are presented to demonstrate the performance of our proposed methods. To the best of our knowledge, no prior studies have been conducted to investigate accelerated first-order methods with complexity guarantees for convex optimization with LLCG. All the complexity results obtained in this paper are new.A systematic approach to Lyapunov analyses of continuous-time models in convex optimizationhttps://zbmath.org/1522.901022023-12-07T16:00:11.105023Z"Moucer, Céline"https://zbmath.org/authors/?q=ai:moucer.celine"Taylor, Adrien"https://zbmath.org/authors/?q=ai:taylor.adrien-b"Bach, Francis"https://zbmath.org/authors/?q=ai:bach.francis-rSummary: First-order methods are often analyzed via their continuous-time models, where their worst-case convergence properties are usually approached via Lyapunov functions. In this work, we provide a systematic and principled approach to finding and verifying Lyapunov functions for classes of ordinary and stochastic differential equations. More precisely, we extend the performance estimation framework, originally proposed by \textit{Y. Drori} and \textit{M. Teboulle} [Math. Program. 145, No. 1--2 (A), 451--482 (2014; Zbl 1300.90068)], to continuous-time models. We retrieve convergence results comparable to those of discrete-time methods using fewer assumptions and inequalities and provide new results for a family of stochastic accelerated gradient flows.Affine invariant convergence rates of the conditional gradient methodhttps://zbmath.org/1522.901032023-12-07T16:00:11.105023Z"Peña, Javier F."https://zbmath.org/authors/?q=ai:pena.javier-fSummary: We show that the conditional gradient method for the convex composite problem \(\min_x\{f(x)+\Psi(x)\}\) generates primal and dual iterates with a duality gap converging to zero provided a suitable \textit{growth property} holds and the algorithm makes a judicious choice of stepsizes. The rate of convergence of the duality gap to zero ranges from sublinear to linear depending on the degree of the growth property. The growth property and convergence results depend on the pair \((f,\Psi)\) in an affine invariant and norm-independent fashion.Density and genericity of well-posed vector optimization problemshttps://zbmath.org/1522.901042023-12-07T16:00:11.105023Z"Rocca, Matteo"https://zbmath.org/authors/?q=ai:rocca.matteoSummary: We consider well-posedness properties of vector optimization problems with objective function \(f:X\to Y\) where \(X\) and \(Y\) are Banach spaces and \(Y\) is partially ordered by a closed convex pointed cone with nonempty interior. The vector well-posedness notion considered in this paper is the one due to \textit{D. Dentcheva} and \textit{S. Helbig} [J. Optim. Theory Appl. 89, No. 2, 325--349 (1996; Zbl 0853.90101)], which is a natural extension of Tykhonov well-posedness for scalar optimization problems. When a scalar optimization problem is considered it is possible to prove (see e.g. [\textit{R. Lucchetti}, Convexity and well-posed problems. New York, NY: Springer (2006; Zbl 1106.49001); \textit{A. J. Zaslavski}, Optimization on metric and normed spaces. New York, NY: Springer (2010; Zbl 1225.90004)]) that under some assumptions the set of functions for which the related optimization problem is Tykhonov well-posed is dense or even more is ``big'' i.e. contains a dense \(G_\delta\) set (these results are called genericity results). The aim of this paper is to extend these genericity results to vector optimization problems.An elementary approach to tight worst case complexity analysis of gradient based methodshttps://zbmath.org/1522.901052023-12-07T16:00:11.105023Z"Teboulle, Marc"https://zbmath.org/authors/?q=ai:teboulle.marc"Vaisbourd, Yakov"https://zbmath.org/authors/?q=ai:vaisbourd.yakovSummary: This work presents a novel analysis that allows to achieve tight complexity bounds of gradient-based methods for convex optimization. We start by identifying some of the pitfalls rooted in the classical complexity analysis of the gradient descent method, and show how they can be remedied. Our methodology hinges on elementary and direct arguments in the spirit of the classical analysis. It allows us to establish some new (and reproduce known) tight complexity results for several fundamental algorithms including, gradient descent, proximal point and proximal gradient methods which previously could be proven only through computer-assisted convergence proof arguments.Convergence of an asynchronous block-coordinate forward-backward algorithm for convex composite optimizationhttps://zbmath.org/1522.901062023-12-07T16:00:11.105023Z"Traoré, Cheik"https://zbmath.org/authors/?q=ai:traore.cheik"Salzo, Saverio"https://zbmath.org/authors/?q=ai:salzo.saverio"Villa, Silvia"https://zbmath.org/authors/?q=ai:villa.silviaSummary: In this paper, we study the convergence properties of a randomized block-coordinate descent algorithm for the minimization of a composite convex objective function, where the block-coordinates are updated asynchronously and randomly according to an arbitrary probability distribution. We prove that the iterates generated by the algorithm form a stochastic quasi-Fejér sequence and thus converge almost surely to a minimizer of the objective function. Moreover, we prove a general sublinear rate of convergence in expectation for the function values and a linear rate of convergence in expectation under an error bound condition of Tseng type. Under the same condition strong convergence of the iterates is provided as well as their linear convergence rate.Algorithms to solve unbounded convex vector optimization problemshttps://zbmath.org/1522.901072023-12-07T16:00:11.105023Z"Wagner, Andrea"https://zbmath.org/authors/?q=ai:wagner.andrea"Ulus, Firdevs"https://zbmath.org/authors/?q=ai:ulus.firdevs"Rudloff, Birgit"https://zbmath.org/authors/?q=ai:rudloff.birgit"Kováčová, Gabriela"https://zbmath.org/authors/?q=ai:kovacova.gabriela"Hey, Niklas"https://zbmath.org/authors/?q=ai:hey.niklasSummary: This paper is concerned with solution algorithms for general convex vector optimization problems (CVOPs). So far, solution concepts and approximation algorithms for solving CVOPs exist only for bounded problems [\textit{Ç. Ararat} et al., J. Optim. Theory Appl. 194, No. 2, 681--712 (2022; Zbl 1495.90171)], [\textit{D. Dörfler} et al., Optim. Methods Softw. 37, No. 3, 1006--1026 (2022; Zbl 1502.90164)], [\textit{A. Löhne} et al., J. Glob. Optim. 60, No. 4, 713--736 (2014; Zbl 1334.90160)]. They provide a polyhedral inner and outer approximation of the upper image that have a Hausdorff distance of at most \(\varepsilon\). However, it is well known (see [\textit{F. Ulus}, J. Glob. Optim. 72, No. 4, 731--742 (2018; Zbl 1404.90125)]), that for some unbounded problems such polyhedral approximations do not exist. In this paper, we will propose a generalized solution concept, called an \((\varepsilon,\delta)\)-solution, that allows one to also consider unbounded CVOPs. It is based on additionally bounding the recession cones of the inner and outer polyhedral approximations of the upper image in a meaningful way. An algorithm is proposed that computes such \(\delta\)-outer and \(\delta\)-inner approximations of the recession cone of the upper image. In combination with the results of [Löhne et al., loc. cit.] this provides a primal and a dual algorithm that allow one to compute \((\varepsilon,\delta)\)-solutions of (potentially unbounded) CVOPs. Numerical examples are provided.A dual semismooth Newton based augmented Lagrangian method for large-scale linearly constrained sparse group square-root Lasso problemshttps://zbmath.org/1522.901082023-12-07T16:00:11.105023Z"Wang, Chengjing"https://zbmath.org/authors/?q=ai:wang.chengjing"Tang, Peipei"https://zbmath.org/authors/?q=ai:tang.peipeiSummary: Square-root Lasso problems have already be shown to be robust regression problems. Furthermore, square-root regression problems with structured sparsity also plays an important role in statistics and machine learning. In this paper, we focus on the numerical computation of large-scale linearly constrained sparse group square-root Lasso problems. In order to overcome the difficulty that there are two nonsmooth terms in the objective function, we propose a dual semismooth Newton (SSN) based augmented Lagrangian method (ALM) for it. That is, we apply the ALM to the dual problem with the subproblem solved by the SSN method. To apply the SSN method, the positive definiteness of the generalized Jacobian is very important. Hence we characterize the equivalence of its positive definiteness and the constraint nondegeneracy condition of the corresponding primal problem. In numerical implementation, we fully employ the second order sparsity so that the Newton direction can be efficiently obtained. Numerical experiments demonstrate the efficiency of the proposed algorithm.Stochastic saddle point problems with decision-dependent distributionshttps://zbmath.org/1522.901092023-12-07T16:00:11.105023Z"Wood, Killian"https://zbmath.org/authors/?q=ai:wood.killian"Dall'Anese, Emiliano"https://zbmath.org/authors/?q=ai:dallanese.emilianoSummary: This paper focuses on stochastic saddle point problems with decision-dependent distributions. These are problems whose objective is the expected value of a stochastic payoff function and whose data distribution drifts in response to decision variables -- a phenomenon represented by a distributional map. A common approach to accommodating distributional shift is to retrain optimal decisions once a new distribution is revealed, or repeated retraining. We introduce the notion of equilibrium points, which are the fixed points of this repeated retraining procedure, and provide sufficient conditions for their existence and uniqueness. To find equilibrium points, we develop deterministic and stochastic primal-dual algorithms and demonstrate their convergence with constant step size in the former and polynomial decay step-size schedule in the latter. By modeling errors emerging from a stochastic gradient estimator as sub-Weibull random variables, we provide error bounds in expectation and in high probability that hold for each iteration. Without additional knowledge of the distributional map, computing saddle points is intractable. Thus we propose a condition on the distributional map -- which we call opposing mixture dominance -- that ensures that the objective is strongly-convex-strongly-concave. Finally, we demonstrate that derivative-free algorithms with a single function evaluation are capable of approximating saddle points.Online bandit convex optimisation with stochastic constraints via two-point feedbackhttps://zbmath.org/1522.901102023-12-07T16:00:11.105023Z"Yu, Jichi"https://zbmath.org/authors/?q=ai:yu.jichi"Li, Jueyou"https://zbmath.org/authors/?q=ai:li.jueyou"Chen, Guo"https://zbmath.org/authors/?q=ai:chen.guoSummary: In this paper, an online convex optimisation problem with stochastic constraints in the bandit setup is investigated. We are particularly interested in the scenario where the gradient information of both loss and constraint functions is unavailable. Under this scenario, only the values of loss and constraint functions at a few random points near the decision are provided to the decision maker after the decision is submitted. We first propose an online bandit algorithm based on the virtual queue in which two-point feedback is used to approximate the gradient feedback. Then we adopt the static benchmark to analyse the optimisation performance and establish the sub-linear expected static regret and sub-linear expected constraint violations of the proposed algorithm in the two-point bandit feedback setup. Moreover, the expected static regret and constraint violations are further improved to \(\mathcal{O}(\ln T)\) when loss functions satisfy the condition of strong convexity. Finally, an online job scheduling numerical simulation is shown to demonstrate the performance of the proposed method and to corroborate the theoretical guarantees.Superiorization under a growth condition on an objective functionhttps://zbmath.org/1522.901112023-12-07T16:00:11.105023Z"Zaslavski, Alexander J."https://zbmath.org/authors/?q=ai:zaslavski.alexander-jSummary: We study a constrained minimization problem with a convex objective function and with a feasible region, which is the intersection of a finite family of closed convex constraint sets. We assume that an objective function satisfies a growth condition and use a projected subgradient method combined with a dynamic string-averaging projection method, with variable strings and variable weights, as a feasibility-seeking algorithm. It is shown that any sequence, generated by the superiorized version of a dynamic string-averaging projection algorithm, not only converges to a feasible point but, additionally, also either its limit point solves the constrained minimization problem or the sequence is strictly Fejér monotone with respect to the solution set.A unified primal-dual algorithm framework for inequality constrained problemshttps://zbmath.org/1522.901122023-12-07T16:00:11.105023Z"Zhu, Zhenyuan"https://zbmath.org/authors/?q=ai:zhu.zhenyuan"Chen, Fan"https://zbmath.org/authors/?q=ai:chen.fan"Zhang, Junyu"https://zbmath.org/authors/?q=ai:zhang.junyu"Wen, Zaiwen"https://zbmath.org/authors/?q=ai:wen.zaiwenSummary: In this paper, we propose a unified primal-dual algorithm framework based on the augmented Lagrangian function for composite convex problems with conic inequality constraints. The new framework is highly versatile. First, it not only covers many existing algorithms such as PDHG, Chambolle-Pock, GDA, OGDA and linearized ALM, but also guides us to design a new efficient algorithm called Semi-OGDA (SOGDA). Second, it enables us to study the role of the augmented penalty term in the convergence analysis. Interestingly, a properly selected penalty not only improves the numerical performance of the above methods, but also theoretically enables the convergence of algorithms like PDHG and SOGDA. Under properly designed step sizes and penalty term, our unified framework preserves the \(\mathcal{O}(1/N)\) ergodic convergence while not requiring any prior knowledge about the magnitude of the optimal Lagrangian multiplier. Linear convergence rate for affine equality constrained problem is also obtained given appropriate conditions. Finally, numerical experiments on linear programming, \(\ell_1\) minimization problem, and multi-block basis pursuit problem demonstrate the efficiency of our methods.A proximal trust-region method for nonsmooth optimization with inexact function and gradient evaluationshttps://zbmath.org/1522.901132023-12-07T16:00:11.105023Z"Baraldi, Robert J."https://zbmath.org/authors/?q=ai:baraldi.robert-j"Kouri, Drew P."https://zbmath.org/authors/?q=ai:kouri.drew-pSummary: Many applications require minimizing the sum of smooth and nonsmooth functions. For example, basis pursuit denoising problems in data science require minimizing a measure of data misfit plus an \(\ell^1\)-regularizer. Similar problems arise in the optimal control of partial differential equations (PDEs) when sparsity of the control is desired. We develop a novel trust-region method to minimize the sum of a smooth nonconvex function and a nonsmooth convex function. Our method is unique in that it permits and systematically controls the use of inexact objective function and derivative evaluations. When using a quadratic Taylor model for the trust-region subproblem, our algorithm is an inexact, matrix-free proximal Newton-type method that permits indefinite Hessians. We prove global convergence of our method in Hilbert space and demonstrate its efficacy on three examples from data science and PDE-constrained optimization.Computational study of local search methods for a D.C. optimization problem with inequality constraintshttps://zbmath.org/1522.901142023-12-07T16:00:11.105023Z"Barkova, M. V."https://zbmath.org/authors/?q=ai:barkova.maria-v"Strekalovskiy, A. S."https://zbmath.org/authors/?q=ai:strekalovskii.aleksander-sergeevichSummary: This paper addresses a nonconvex optimization problem where the cost function and inequality constraints are d.c. functions. Two special local search methods based on the idea of the consecutive solution of partially linearized problems are developed. The latter problems turn out to be convex and therefore solvable with the help of software packages for convex optimization. The first method linearizes both the objective function of the problem and all constraints functions. The second approach is based on the reduction of the original problem to a penalized problem without constraints via the exact penalization theory. The methods developed were computationally tested on some well-known test examples and specially generated problems with known local and global solutions.
For the entire collection see [Zbl 1508.90001].A dynamic smoothing technique for a class of nonsmooth optimization problems on manifoldshttps://zbmath.org/1522.901152023-12-07T16:00:11.105023Z"Beck, Amir"https://zbmath.org/authors/?q=ai:beck.amir"Rosset, Israel"https://zbmath.org/authors/?q=ai:rosset.israelSummary: We consider the problem of minimizing the sum of a smooth nonconvex function and a nonsmooth convex function over a compact embedded submanifold. We describe an algorithm, which we refer to as ``dynamic smoothing gradient descent on manifolds'' (DSGM), that is based on applying Riemmanian gradient steps on a series of smooth approximations of the objective function that are determined by a diminishing sequence of smoothing parameters. The DSGM algorithm is simple and can be easily employed for a broad class of problems without any complex adjustments. We show that all accumulation points of the sequence generated by the method are stationary. We devise a convergence rate of \(O(\frac{1}{k^{1/3}})\) in terms of an optimality measure that can be easily computed. Numerical experiments illustrate the potential of the DSGM method.Constraint qualification with Schauder basis for infinite programming problemshttps://zbmath.org/1522.901162023-12-07T16:00:11.105023Z"Bednarczuk, E. M."https://zbmath.org/authors/?q=ai:bednarczuk.ewa-m"Leśniewski, K. W."https://zbmath.org/authors/?q=ai:lesniewski.k-w"Rutkowski, K. E."https://zbmath.org/authors/?q=ai:rutkowski.krzysztof-eSummary: We consider infinite programming problems with constraint sets defined by systems of infinite number of inequalities and equations given by continuously differentiable functions defined on Banach spaces. In the approach proposed here we represent these systems with the help of coefficients in a given Schauder basis. We prove Abadie constraint qualification under the new infinite-dimensional Relaxed Constant Rank Constraint Qualification Plus and we discuss the existence of Lagrange multipliers via Hurwicz set. The main tools are: Rank Theorem and Lyusternik-Graves theorem.On the sum rule for the weak subdiferential and some properties of augmented normal coneshttps://zbmath.org/1522.901172023-12-07T16:00:11.105023Z"Bila, Samet"https://zbmath.org/authors/?q=ai:bila.samet"Kasimbeyli, Refail"https://zbmath.org/authors/?q=ai:kasimbeyli.refail"Farajzadeh, Ali"https://zbmath.org/authors/?q=ai:farajzadeh.ali-pSummary: This paper investigates the sum rule for the weak subdifferential. It is shown that the weak subdifferentials of some classes of the Clarke directionally differentiable functions and the tangentially convex functions, satisfy this property in the form of equality. Moreover we show that the ``sup relation'' for the weak subdifferential, can be formulated as a ``max relation'' for the mentioned classes of functions. In this work, we also study a relation between the weak subdifferential of the indicator function and the augmented normal cone to a nonconvex set, which is used to establish some additional properties of nonconvex sets.The minimization of piecewise functions: pseudo stationarityhttps://zbmath.org/1522.901182023-12-07T16:00:11.105023Z"Cui, Ying"https://zbmath.org/authors/?q=ai:cui.ying"Liu, Junyi"https://zbmath.org/authors/?q=ai:liu.junyi"Pang, Jong-Shi"https://zbmath.org/authors/?q=ai:pang.jong-shiSummary: There are many significant applied contexts that require the solution of discontinuous optimization problems in finite dimensions. Yet these problems are very difficult, both computationally and analytically. With the functions being discontinuous and a minimizer (local or global) of the problems, even if it exists, being impossible to verifiably compute, a foremost question is what kind of ``stationary solutions'' one can expect to obtain; these solutions provide promising candidates for minimizers; i.e., their defining conditions are necessary for optimality. Motivated by recent results on sparse optimization, we introduce in this paper such a kind of solution, termed ``pseudo B- (for Bouligand) stationary solution'', for a broad class of discontinuous optimization problems with objective and constraint defined by indicator functions of the positive real axis composite with functions that are possibly nonsmooth. We present two approaches for computing such a solution. One approach is based on lifting the problem to a higher dimension via the epigraphical formulation of the indicator functions; this requires the addition of some auxiliary variables. The other approach is based on certain continuous (albeit not necessarily differentiable) piecewise approximations of the indicator functions and the convergence to a pseudo B-stationary solution of the original problem is established. The conditions for convergence are discussed and illustrated by an example.Worst-case complexity of TRACE with inexact subproblem solutions for nonconvex smooth optimizationhttps://zbmath.org/1522.901192023-12-07T16:00:11.105023Z"Curtis, Frank E."https://zbmath.org/authors/?q=ai:curtis.frank-e"Wang, Qi"https://zbmath.org/authors/?q=ai:wang.qi.10|wang.qi.4|wang.qi.9|wang.qi.31|wang.qi.2|wang.qi.29|wang.qi.26|wang.qi.12|wang.qi|wang.qi.7|wang.qi.8|wang.qi.1|wang.qi.27|wang.qi.6Summary: An algorithm for solving nonconvex smooth optimization problems is proposed, analyzed, and tested. The algorithm is an extension of the trust-region algorithm with contractions and expansions (TRACE) [\textit{F. E. Curtis} et al., Math. Program. 162, No. 1--2 (A), 1--32 (2017; Zbl 1360.49020)]. In particular, the extension allows the algorithm to use inexact solutions of the arising subproblems, which is an important feature for solving large-scale problems. Inexactness is allowed in a manner such that the optimal iteration complexity of \(\mathcal{O}(\epsilon^{-3/2})\) for attaining an \(\epsilon\)-approximate first-order stationary point is maintained, while the worst-case complexity in terms of Hessian-vector products may be significantly improved as compared to the original TRACE. Numerical experiments show the benefits of allowing inexact subproblem solutions and that the algorithm compares favorably to state-of-the-art techniques.Global optimization via Schrödinger-Föllmer diffusionhttps://zbmath.org/1522.901202023-12-07T16:00:11.105023Z"Dai, Yin"https://zbmath.org/authors/?q=ai:dai.yin"Jiao, Yuling"https://zbmath.org/authors/?q=ai:jiao.yuling"Kang, Lican"https://zbmath.org/authors/?q=ai:kang.lican"Lu, Xiliang"https://zbmath.org/authors/?q=ai:lu.xiliang"Yang, Jerry Zhijian"https://zbmath.org/authors/?q=ai:yang.jerry-zhijianSummary: We study the problem of approximately finding global minimizers of \(V(x):\mathbb{R}^d\rightarrow \mathbb{R}\) via sampling from a probability distribution \(\mu_{\sigma}\) with density \(p_{\sigma}(x)=\frac{\exp(-V(x)/\sigma)}{\int_{\mathbb{R}^d}\exp(-V(y)/\sigma)dy}\) with respect to the Lebesgue measure for \(\sigma\in(0,1]\) small enough. We analyze a sampler based on the Euler-Maruyama discretization of the Schrödinger-Föllmer diffusion processes with stochastic approximation under appropriate assumptions on the step size \(s\) and the potential \(V\). We prove that the output of the proposed sampler is an approximate global minimizer of \(V(x)\) with high probability at cost of sampling \(\mathcal{O}(d^3)\) standard normal random variables. Numerical studies illustrate the effectiveness of the proposed method and its superiority to the Langevin method.Regularizing orientation estimation in cryogenic electron microscopy three-dimensional map refinement through measure-based lifting over Riemannian manifoldshttps://zbmath.org/1522.901212023-12-07T16:00:11.105023Z"Diepeveen, Willem"https://zbmath.org/authors/?q=ai:diepeveen.willem"Lellmann, Jan"https://zbmath.org/authors/?q=ai:lellmann.jan"Öktem, Ozan"https://zbmath.org/authors/?q=ai:oktem.ozan"Schönlieb, Carola-Bibiane"https://zbmath.org/authors/?q=ai:schonlieb.carola-bibianeSummary: Motivated by the trade-off between noise robustness and data consistency for joint three-imensional (3D) map reconstruction and rotation estimation in single particle cryogenic-electron microscopy (Cryo-EM), we propose ellipsoidal support lifting (ESL), a measure-based lifting scheme for regularizing and approximating the global minimizer of a smooth function over a Riemannian manifold. Under a uniqueness assumption on the minimizer we show several theoretical results, in particular well-posedness of the method and an error bound due to the induced bias with respect to the global minimizer. Additionally, we use the developed theory to integrate the measure-based lifting scheme into an alternating update method for joint homogeneous 3D map reconstruction and rotation estimation, where typically tens of thousands of manifold-valued minimization problems have to be solved and where regularization is necessary because of the high noise levels in the data. The joint recovery method is used to test both the theoretical predictions and algorithmic performance through numerical experiments with Cryo-EM data. In particular, the induced bias due to the regularizing effect of ESL empirically estimates better rotations, i.e., rotations closer to the ground truth, than global optimization would.Steering exact penalty DCA for nonsmooth DC optimisation problems with equality and inequality constraintshttps://zbmath.org/1522.901222023-12-07T16:00:11.105023Z"Dolgopolik, M. V."https://zbmath.org/authors/?q=ai:dolgopolik.maxim-vladimirovichSummary: We propose and study a version of the DCA (Difference-of-Convex functions Algorithm) using the \(\ell_1\) penalty function for solving nonsmooth DC optimisation problems with nonsmooth DC equality and inequality constraints. The method employs an adaptive penalty updating strategy to improve its performance. This strategy is based on the so-called steering exact penalty methodology and relies on solving some auxiliary convex subproblems to determine a suitable value of the penalty parameter. We present a detailed convergence analysis of the method and illustrate its practical performance by applying the method to two nonsmooth discrete optimal control problem.Pareto epsilon-subdifferential sum rule for set-valued mappings and applications to set optimizationhttps://zbmath.org/1522.901232023-12-07T16:00:11.105023Z"Echchaabaoui, El Mahjoub"https://zbmath.org/authors/?q=ai:echchaabaoui.el-mahjoub"Laghdir, Mohamed"https://zbmath.org/authors/?q=ai:laghdir.mohamedSummary: In this paper, we are mainly concerned with a rule for efficient (Pareto) approximate subdifferential, concerning the sum of two cone-convex set-valued vector mappings, taking values in finite or infinite-dimensional preordred spaces. The obtained formula is exact and holds under the connectedness or Attouch-Brézis qualification conditions and the regular subdifferentiability. This formula is applied to establish approximate necessary and sufficient optimality conditions for the existence of the approximate Pareto (weak or proper) efficient solutions of a set-valued vector optimization problem.An alternating structure-adapted Bregman proximal gradient descent algorithm for constrained nonconvex nonsmooth optimization problems and its inertial varianthttps://zbmath.org/1522.901242023-12-07T16:00:11.105023Z"Gao, Xue"https://zbmath.org/authors/?q=ai:gao.xue"Cai, Xingju"https://zbmath.org/authors/?q=ai:cai.xingju"Wang, Xiangfeng"https://zbmath.org/authors/?q=ai:wang.xiangfeng"Han, Deren"https://zbmath.org/authors/?q=ai:han.derenSummary: We consider the nonconvex nonsmooth minimization problem over abstract sets, whose objective function is the sum of a proper lower semicontinuous biconvex function of the entire variables and two smooth nonconvex functions of their private variables. Fully exploiting the problem structure, we propose an alternating structure-adapted Bregman proximal (ASABP for short) gradient descent algorithm, where the geometry of the abstract set and the function is captured by employing generalized Bregman function. Under the assumption that the underlying function satisfies the Kurdyka-Łojasiewicz property, we prove that each bounded sequence generated by ASABP globally converges to a critical point. We then adopt an inertial strategy to accelerate the ASABP algorithm (IASABP), and utilize a backtracking line search scheme to find ``suitable'' step sizes, making the algorithm efficient and robust. The global \(O(1/K)\) sublinear convergence rate measured by Bregman distance is also established. Furthermore, to illustrate the potential of ASABP and its inertial version (IASABP), we apply them to solving the Poisson linear inverse problem, and the results are promising.Sublinear scalarizations for proper and approximate proper efficient points in nonconvex vector optimizationhttps://zbmath.org/1522.901252023-12-07T16:00:11.105023Z"García-Castaño, Fernando"https://zbmath.org/authors/?q=ai:garcia-castano.fernando"Melguizo-Padial, Miguel Ángel"https://zbmath.org/authors/?q=ai:padial.miguel-angel-melguizo"Parzanese, G."https://zbmath.org/authors/?q=ai:parzanese.gSummary: We show that under a separation property, a \(\mathcal{Q} \)-minimal point in a normed space is the minimum of a given sublinear function. This fact provides sufficient conditions, via scalarization, for nine types of proper efficient points; establishing a characterization in the particular case of Benson proper efficient points. We also obtain necessary and sufficient conditions in terms of scalarization for approximate Benson and Henig proper efficient points. The separation property we handle is a variation of another known property and our scalarization results do not require convexity or boundedness assumptions.Worst-case evaluation complexity of a quadratic penalty method for nonconvex optimizationhttps://zbmath.org/1522.901262023-12-07T16:00:11.105023Z"Grapiglia, Geovani Nunes"https://zbmath.org/authors/?q=ai:grapiglia.geovani-nunesSummary: This paper addresses the worst-case evaluation complexity of a version of the standard quadratic penalty method for smooth nonconvex optimization problems with constraints. The method analysed allows inexact solution of the subproblems and do not require prior knowledge of the Lipschitz constants related with the problem. When an approximate feasible point is used as starting point, it is shown that the referred method takes at most \(\mathcal{O} \left( \log (\sigma_0^{-1} \epsilon^{-2}) \right)\) outer iterations to generate an \(\epsilon\)-approximate KKT point, where \(\sigma_0\) is the first penalty parameter. For equality constrained problems, this bound yields to an evaluation complexity bound of \(\mathcal{O}(\epsilon^{-4})\), when \(\sigma_0 = \epsilon^{-2}\) and suitable first-order methods are used as inner solvers. For problems having only linear equality constraints, an evaluation complexity bound of \(\mathcal{O}(\epsilon^{-(p+1)/p})\) is established when appropriate \(p\)-order methods \((p \geq 2)\) are used as inner solvers. Illustrative numerical results are also presented and corroborate the theoretical predictions.The convergence properties of infeasible inexact proximal alternating linearized minimizationhttps://zbmath.org/1522.901272023-12-07T16:00:11.105023Z"Hu, Yukuan"https://zbmath.org/authors/?q=ai:hu.yukuan"Liu, Xin"https://zbmath.org/authors/?q=ai:liu.xin.1Summary: The proximal alternating linearized minimization (PALM) method suits well for solving block-structured optimization problems, which are ubiquitous in real applications. In the cases where subproblems do not have closed-form solutions, e.g., due to complex constraints, infeasible subsolvers are indispensable, giving rise to an infeasible inexact PALM (PALM-I). Numerous efforts have been devoted to analyzing the feasible PALM, while little attention has been paid to the PALM-I. The usage of the PALM-I thus lacks a theoretical guarantee. The essential difficulty of analysis consists in the objective value nonmonotonicity induced by the infeasibility. We study in the present work the convergence properties of the PALM-I. In particular, we construct a surrogate sequence to surmount the nonmonotonicity issue and devise an implementable inexact criterion. Based upon these, we manage to establish the stationarity of any accumulation point, and moreover, show the iterate convergence and the asymptotic convergence rates under the assumption of the Łojasiewicz property. The prominent advantages of the PALM-I on CPU time are illustrated via numerical experiments on problems arising from quantum physics and 3-dimensional anisotropic frictional contact.Certifying the absence of spurious local minima at infinityhttps://zbmath.org/1522.901282023-12-07T16:00:11.105023Z"Josz, Cédric"https://zbmath.org/authors/?q=ai:josz.cedric"Li, Xiaopeng"https://zbmath.org/authors/?q=ai:li.xiaopengSummary: When searching for global optima of nonconvex unconstrained optimization problems, it is desirable that every local minimum be a global minimum. This property of having no spurious local minima is true in various problems of interest nowadays, including principal component analysis, matrix sensing, and linear neural networks. However, since these problems are noncoercive, they may yet have spurious local minima at infinity. The classical tools used to analyze the optimization landscape, namely the gradient and the Hessian, are incapable of detecting spurious local minima at infinity. In this paper, we identify conditions that certify the absence of spurious local minima at infinity, one of which is having bounded subgradient trajectories. We check that they hold in several applications of interest.A derivative-free nonlinear least squares solverhttps://zbmath.org/1522.901292023-12-07T16:00:11.105023Z"Kaporin, Igor"https://zbmath.org/authors/?q=ai:kaporin.igor-eSummary: A nonlinear least squares iterative solver developed earlier by the author is modified to fit the derivative-free optimization paradigm. The proposed algorithm is based on easily parallelizable computational kernels such as small dense matrix factorizations and elementary vector operations and therefore has a potential for a quite efficient implementation on modern high-performance computers. Numerical results are presented for several standard test problems to demonstrate the competitiveness of the proposed method.
For the entire collection see [Zbl 1508.90001].Cardinality-constrained optimization problems in general position and beyondhttps://zbmath.org/1522.901302023-12-07T16:00:11.105023Z"Lämmel, Sebastian"https://zbmath.org/authors/?q=ai:lammel.sebastian"Shikhman, Vladimir"https://zbmath.org/authors/?q=ai:shikhman.vladimirSummary: We study cardinality-constrained optimization problems (CCOP) in general position, i.e. those optimization-related properties that are fulfilled for a dense and open subset of their defining functions. We show that the well-known cardinality-constrained linear independence constraint qualification (CC-LICQ) is generic in this sense. For M-stationary points we define nondegeneracy and show that it is a generic property too. In particular, the sparsity constraint turns out to be active at all minimizers of a generíc CCOP. Moreover, we describe the global structure of CCOP in the sense of Morse theory, emphasizing the strength of the generic approach. Here, we prove that multiple cells need to be attached, each of dimension coinciding with the proposed M-index of nondegenerate M-stationary points. Beyond this generic viewpoint, we study singularities of CCOP. For that, the relation between nondegeneracy and strong stability in the sense of Kojima is examined. We show that nondegeneracy implies the latter, while the reverse implication is in general not true. To fill the gap, we fully characterize the strong stability of M-stationary points under CC-LICQ by first- and second-order information of CCOP defining functions. Finally, we compare nondegeneracy and strong stability of M-stationary points with second-order sufficient conditions recently introduced in the literature.A forward-backward algorithm with different inertial terms for structured non-convex minimization problemshttps://zbmath.org/1522.901312023-12-07T16:00:11.105023Z"László, Szilárd Csaba"https://zbmath.org/authors/?q=ai:laszlo.szilardSummary: We investigate an inertial forward-backward algorithm in connection with the minimization of the sum of a non-smooth and possibly non-convex and a non-convex differentiable function. The algorithm is formulated in the spirit of the famous FISTA method; however, the setting is non-convex and we allow different inertial terms. Moreover, the inertial parameters in our algorithm can take negative values too. We also treat the case when the non-smooth function is convex, and we show that in this case a better step size can be allowed. Further, we show that our numerical schemes can successfully be used in DC-programming. We prove some abstract convergence results which applied to our numerical schemes allow us to show that the generated sequences converge to a critical point of the objective function, provided a regularization of the objective function satisfies the Kurdyka-Łojasiewicz property. Further, we obtain a general result that applied to our numerical schemes ensures convergence rates for the generated sequences and for the objective function values formulated in terms of the KL exponent of a regularization of the objective function. Finally, we apply our results to image restoration.On solving difference of convex functions programs with linear complementarity constraintshttps://zbmath.org/1522.901322023-12-07T16:00:11.105023Z"Le Thi, Hoai An"https://zbmath.org/authors/?q=ai:le-thi-hoai-an."Nguyen, Thi Minh Tam"https://zbmath.org/authors/?q=ai:nguyen.thi-minh-tam"Dinh, Tao Pham"https://zbmath.org/authors/?q=ai:pham-dinh-tao.Summary: We address a large class of Mathematical Programs with Linear Complementarity Constraints which minimizes a continuously differentiable DC function (Difference of Convex functions) on a set defined by linear constraints and linear complementarity constraints, named Difference of Convex functions programs with Linear Complementarity Constraints. Using exact penalty techniques, we reformulate it, via four penalty functions, as standard Difference of Convex functions programs. The difference of convex functions algorithm (DCA), an efficient approach in nonconvex programming framework, is then developed to solve the resulting problems. Two particular cases are considered: quadratic problems with linear complementarity constraints and asymmetric eigenvalue complementarity problems. Numerical experiments are performed on several benchmark data, and the results show the effectiveness and the superiority of the proposed approaches comparing with some standard methods.Average curvature FISTA for nonconvex smooth composite optimization problemshttps://zbmath.org/1522.901332023-12-07T16:00:11.105023Z"Liang, Jiaming"https://zbmath.org/authors/?q=ai:liang.jiaming"Monteiro, Renato D. C."https://zbmath.org/authors/?q=ai:monteiro.renato-d-cSummary: A previous authors' paper introduces an accelerated composite gradient (ACG) variant, namely AC-ACG, for solving nonconvex smooth composite optimization (N-SCO) problems. In contrast to other ACG variants, AC-ACG estimates the local upper curvature of the N-SCO problem by using the average of the observed upper-Lipschitz curvatures obtained during the previous iterations, and uses this estimation and two composite resolvent evaluations to compute the next iterate. This paper presents an alternative FISTA-type ACG variant, namely AC-FISTA, which has the following additional features: (i) it performs an average of one composite resolvent evaluation per iteration; and (ii) it estimates the local upper curvature by using the average of the previously observed upper (instead of upper-Lipschitz) curvatures. These two properties acting together yield a practical AC-FISTA variant which substantially outperforms earlier ACG variants, including the AC-ACG variants discussed in the aforementioned authors' paper.Normal cones intersection rule and optimality analysis for low-rank matrix optimization with affine manifoldshttps://zbmath.org/1522.901342023-12-07T16:00:11.105023Z"Li, Xinrong"https://zbmath.org/authors/?q=ai:li.xinrong"Luo, Ziyan"https://zbmath.org/authors/?q=ai:luo.ziyanSummary: The low-rank matrix optimization with affine manifold (rank-MOA) aims to minimize a continuously differentiable function over a low-rank set intersecting with an affine manifold. This paper is devoted to the optimality analysis for rank-MOA. As a cornerstone, the intersection rule of the Fréchet normal cone to the feasible set of rank-MOA is established under some mild linear independence assumptions. Aided with the resulting explicit formulae of the underlying normal cones, the so-called \(F\)-stationary point and the \(\alpha\)-stationary point of rank-MOA are investigated and the relationship with local/global minimizers are then revealed in terms of first-order optimality conditions. Furthermore, the second-order optimality analysis, including the necessary and sufficient conditions, is proposed based on the second-order differentiation information of the model. All these results will enrich the theory of low-rank matrix optimization and give potential clues to designing efficient numerical algorithms for seeking low-rank solutions. Meanwhile, two specific applications of rank-MOA are discussed to illustrate our proposed optimality analysis.Modified subgradient extragradient algorithms with a new line-search rule for variational inequalitieshttps://zbmath.org/1522.901352023-12-07T16:00:11.105023Z"Long, Xian-Jun"https://zbmath.org/authors/?q=ai:long.xianjun"Yang, Jing"https://zbmath.org/authors/?q=ai:yang.jing.4|yang.jing.6|yang.jing.2|yang.jing.11|yang.jing.1|yang.jing.7|yang.jing.5|yang.jing"Cho, Yeol Je"https://zbmath.org/authors/?q=ai:cho.yeol-jeSummary: In this paper, we introduce a modified subgradient extragradient algorithm with a new line-search rule for solving pseudomonotone variational inequalities with non-Lipschitz mappings. The new line-search rule is designed by the golden radio \((\sqrt{5}+1)/2\). We prove the strong convergence theorem under some appropriate conditions in real Hilbert spaces. Finally, we give some numerical experiments to illustrate the performances and advantages of the proposed algorithm.Combined approach with second-order optimality conditions for bilevel programming problemshttps://zbmath.org/1522.901362023-12-07T16:00:11.105023Z"Ma, Xiaoxiao"https://zbmath.org/authors/?q=ai:ma.xiaoxiao"Yao, Wei"https://zbmath.org/authors/?q=ai:yao.wei.1|yao.wei"Ye, Jane J."https://zbmath.org/authors/?q=ai:ye.jane-j"Zhang, Jin"https://zbmath.org/authors/?q=ai:zhang.jin.2Summary: We propose a combined approach with second-order optimality conditions of the lower level problem to study constraint qualifications and optimality conditions for bilevel programming problems. The new method is inspired by the combined approach developed by Ye and Zhu in 2010, where the authors combined the classical first-order and the value function approaches to derive new necessary optimality conditions. In our approach, we add a second-order optimality condition to the combined program as a new constraint. We show that when all known approaches fail, adding the second-order optimality condition as a constraint makes the corresponding partial calmness condition and the resulting necessary optimality condition easier to hold. We also give some discussions on advantages and disadvantages of the combined approaches with the first-order and the second-order information.A regularization of DC optimizationhttps://zbmath.org/1522.901372023-12-07T16:00:11.105023Z"Moudafi, Abdellatif"https://zbmath.org/authors/?q=ai:moudafi.abdellatifSummary: Numerous models of real world nonconvex optimization can be formulated as DC optimization problems which consist in minimizing a difference of two convex functions, see for instance [\textit{Le Thi Hoai An} and \textit{Pham Dinh Tao}, Ann. Oper. Res. 133, 23--46 (2005; Zbl 1116.90122)]. A popular approach to address nonsmooth terms in convex optimization is to approximate them with their Moreau envelopes, see for example [\textit{I. Yamada} et al., Springer Optim. Appl. 49, 345--390 (2011; Zbl 1263.47088)]. In the spirit of an \textit{J. B. Hiriart-Urruty}'s idea [J. Math. Anal. Appl. 162, No. 1, 196--209 (1991; Zbl 0807.46041)], we propose a complete smooth approximation of the original problem that relies on Moreau envelopes with eventually different regularization parameters. This would allow to enforcing the regularization of the convex or the concave part. A parallel proximal algorithm based on the classical gradient descent method is also proposed.An apocalypse-free first-order low-rank optimization algorithm with at most one rank reduction attempt per iterationhttps://zbmath.org/1522.901382023-12-07T16:00:11.105023Z"Olikier, Guillaume"https://zbmath.org/authors/?q=ai:olikier.guillaume"Absil, P.-A."https://zbmath.org/authors/?q=ai:absil.pierre-antoineSummary: We consider the problem of minimizing a differentiable function with locally Lipschitz continuous gradient on the real determinantal variety and present a first-order algorithm designed to find a stationary point of that problem. This algorithm applies steps of a retraction-free descent method proposed by \textit{R. Schneider} and \textit{A. Uschmajew} [SIAM J. Optim. 25, No. 1, 622--646 (2015; Zbl 1355.65079)], while taking the numerical rank into account to attempt rank reductions. We prove that this algorithm produces a sequence of iterates whose accumulation points are stationary and therefore does not follow the so-called apocalypses described by \textit{E. Levin} et al. [Math. Program. 199, No. 1--2 (A), 831--864 (2023; Zbl 07681267)]. Moreover, the rank reduction mechanism of this algorithm requires at most one rank reduction attempt per iteration, in contrast with the one of the P\(^2\)GDR algorithm introduced by \textit{G. Olikier} et al. [``An apocalypse-free first-order low-rank optimization algorithm'', Preprint, \url{arXiv:2201.03962}], which can require a number of rank reduction attempts equal to the rank of the iterate in the worst-case scenario.Local optimality for stationary points of group zero-norm regularized problems and equivalent surrogateshttps://zbmath.org/1522.901392023-12-07T16:00:11.105023Z"Pan, Shaohua"https://zbmath.org/authors/?q=ai:pan.shaohua"Liang, Ling"https://zbmath.org/authors/?q=ai:liang.ling"Liu, Yulan"https://zbmath.org/authors/?q=ai:liu.yulanSummary: This paper focuses on the local optimality for the stationary points of the composite group zero-norm regularized problem and its equivalent surrogates. First, by using the structure of the composite group zero-norm and its second subderivative characterization, we achieve several local optimal conditions for a stationary point of the group zero-norm regularized problem. Then, we obtain a family of equivalent surrogates for the group zero-norm regularized problem from a class of global exact penalties of its MPEC reformulation, established under the calmness of a partial perturbation to the composite group zero-norm constraint system. For the stationary points of these surrogates, we study their local optimality to the surrogates themselves and the group zero-norm regularized problem. The local optimality conditions obtained in this work not only recover the existing ones for zero-norm regularized problems, but also provide new criteria to judge the local optimality of a stationary point yielded by an algorithm for solving the corresponding surrogate problems.Smooth over-parameterized solvers for non-smooth structured optimizationhttps://zbmath.org/1522.901402023-12-07T16:00:11.105023Z"Poon, Clarice"https://zbmath.org/authors/?q=ai:poon.clarice"Peyré, Gabriel"https://zbmath.org/authors/?q=ai:peyre.gabrielSummary: Non-smooth optimization is a core ingredient of many imaging or machine learning pipelines. Non-smoothness encodes structural constraints on the solutions, such as sparsity, group sparsity, low-rank and sharp edges. It is also the basis for the definition of robust loss functions and scale-free functionals such as square-root Lasso. Standard approaches to deal with non-smoothness leverage either proximal splitting or coordinate descent. These approaches are effective but usually require parameter tuning, preconditioning or some sort of support pruning. In this work, we advocate and study a different route, which operates a non-convex but smooth over-parameterization of the underlying non-smooth optimization problems. This generalizes quadratic variational forms that are at the heart of the popular Iterative Reweighted Least Squares. Our main theoretical contribution connects gradient descent on this reformulation to a mirror descent flow with a varying Hessian metric. This analysis is crucial to derive convergence bounds that are dimension-free. This explains the efficiency of the method when using small grid sizes in imaging. Our main algorithmic contribution is to apply the Variable Projection method which defines a new formulation by explicitly minimizing over part of the variables. This leads to a better conditioning of the minimized functional and improves the convergence of simple but very efficient gradient-based methods, for instance quasi-Newton solvers. We exemplify the use of this new solver for the resolution of regularized regression problems for inverse problems and supervised learning, including total variation prior and non-convex regularizers.Stopping rules for gradient methods for non-convex problems with additive noise in gradienthttps://zbmath.org/1522.901412023-12-07T16:00:11.105023Z"Stonyakin, Fedor"https://zbmath.org/authors/?q=ai:stonyakin.fedor-sergeevich"Kuruzov, Ilya"https://zbmath.org/authors/?q=ai:kuruzov.ilya-a"Polyak, Boris"https://zbmath.org/authors/?q=ai:polyak.boris-tSummary: We study the gradient method under the assumption that an additively inexact gradient is available for, generally speaking, non-convex problems. The non-convexity of the objective function, as well as the use of an inexactness specified gradient at iterations, can lead to various problems. For example, the trajectory of the gradient method may be far enough away from the starting point. On the other hand, the unbounded removal of the trajectory of the gradient method in the presence of noise can lead to the removal of the trajectory of the method from the desired global solution. The results of investigating the behavior of the trajectory of the gradient method are obtained under the assumption of the inexactness of the gradient and the condition of gradient dominance. It is well known that such a condition is valid for many important non-convex problems. Moreover, it leads to good complexity guarantees for the gradient method. A rule of early stopping of the gradient method is proposed. Firstly, it guarantees achieving an acceptable quality of the exit point of the method in terms of the function. Secondly, the stopping rule ensures a fairly moderate distance of this point from the chosen initial position. In addition to the gradient method with a constant step, its variant with adaptive step size is also investigated in detail, which makes it possible to apply the developed technique in the case of an unknown Lipschitz constant for the gradient. Some computational experiments have been carried out which demonstrate effectiveness of the proposed stopping rule for the investigated gradient methods.An adaptive superfast inexact proximal augmented Lagrangian method for smooth nonconvex composite optimization problemshttps://zbmath.org/1522.901422023-12-07T16:00:11.105023Z"Sujanani, Arnesh"https://zbmath.org/authors/?q=ai:sujanani.arnesh"Monteiro, Renato D. C."https://zbmath.org/authors/?q=ai:monteiro.renato-d-cSummary: This work presents an adaptive superfast proximal augmented Lagrangian (AS-PAL) method for solving linearly-constrained smooth nonconvex composite optimization problems. Each iteration of AS-PAL inexactly solves a possibly nonconvex proximal augmented Lagrangian (AL) subproblem obtained by an aggressive/adaptive choice of prox stepsize with the aim of substantially improving its computational performance followed by a full Lagrange multiplier update. A major advantage of AS-PAL compared to other AL methods is that it requires no knowledge of parameters (e.g., size of constraint matrix, objective function curvatures, etc) associated with the optimization problem, due to its adaptive nature not only in choosing the prox stepsize but also in using a crucial adaptive accelerated composite gradient variant to solve the proximal AL subproblems. The speed and efficiency of AS-PAL is demonstrated through extensive computational experiments showing that it can solve many instances more than ten times faster than other state-of-the-art penalty and AL methods, particularly when high accuracy is required.A Bregman stochastic method for Nonconvex nonsmooth problem beyond global Lipschitz gradient continuityhttps://zbmath.org/1522.901432023-12-07T16:00:11.105023Z"Wang, Qingsong"https://zbmath.org/authors/?q=ai:wang.qingsong"Han, Deren"https://zbmath.org/authors/?q=ai:han.derenSummary: In this paper, we consider solving a broad class of large-scale nonconvex and nonsmooth minimization problems by a Bregman proximal stochastic gradient (BPSG) algorithm. The objective function of the minimization problem is the composition of a differentiable and a nondifferentiable function, and the differentiable part does not admit a global Lipschitz continuous gradient. Under some suitable conditions, the subsequential convergence of the proposed algorithm is established. And under expectation conditions with the Kurdyka-Łojasiewicz (KL) property, we also prove that the proposed method converges globally. We also apply the BPSG algorithm to solve sparse nonnegative matrix factorization (NMF), symmetric NMF via non-symmetric relaxation, and matrix completion problems under different kernel generating distances, and numerically compare it with other algorithms. The results demonstrate the robustness and effectiveness of the proposed algorithm.Calculus rules of the generalized concave Kurdyka-Łojasiewicz propertyhttps://zbmath.org/1522.901442023-12-07T16:00:11.105023Z"Wang, Xianfu"https://zbmath.org/authors/?q=ai:wang.xianfu"Wang, Ziyuan"https://zbmath.org/authors/?q=ai:wang.ziyuanSummary: In this paper, we propose several calculus rules for the generalized concave Kurdyka-Łojasiewicz (KL) property, which generalize \textit{G. Li} and \textit{T. K. Pong}'s result [Found. Comput. Math. 18, No. 5, 1199--1232 (2018; Zbl 1405.90076)] for KL exponents. The optimal concave desingularizing function has various forms and may be nondifferentiable. Our calculus rules do not assume desingularizing functions to have any specific form nor differentiable, while the known results do. Several examples are also given to show that our calculus rules are applicable to a broader class of functions than the known ones.A regularized Newton method for \(\ell_q\)-norm composite optimization problemshttps://zbmath.org/1522.901452023-12-07T16:00:11.105023Z"Wu, Yuqia"https://zbmath.org/authors/?q=ai:wu.yuqia"Pan, Shaohua"https://zbmath.org/authors/?q=ai:pan.shaohua"Yang, Xiaoqi"https://zbmath.org/authors/?q=ai:yang.xiaoqiSummary: This paper is concerned with \(\ell_q\) \((0<q<1)\)-norm regularized minimization problems with a twice continuously differentiable loss function. For this class of nonconvex and nonsmooth composite problems, many algorithms have been proposed to solve them, most of which are of the first-order type. In this work, we propose a hybrid of the proximal gradient method and the subspace regularized Newton method, called HpgSRN. The whole iterate sequence produced by HpgSRN is proved to have a finite length and to converge to an \(L\)-type stationary point under a mild curve-ratio condition and the Kurdyka-Łojasiewicz property of the cost function; it converges linearly if a further Kurdyka-Łojasiewicz property of exponent \(1/2\) holds. Moreover, a superlinear convergence rate for the iterate sequence is also achieved under an additional local error bound condition. Our convergence results do not require the isolatedness and strict local minimality properties of the \(L\)-stationary point. Numerical comparisons with ZeroFPR, a hybrid of proximal gradient method and quasi-Newton method for the forward-backward envelope of the cost function, proposed in [\textit{A. Themelis} et al., SIAM J. Optim. 28, No. 3, 2274--2303 (2018; Zbl 1404.90106)] for the \(\ell_q\)-norm regularized linear and logistic regressions on real data, indicate that HpgSRN not only requires much less computing time but also yields comparable or even better sparsities and objective function values.Sparse regularization with the \(\ell_0\) normhttps://zbmath.org/1522.901462023-12-07T16:00:11.105023Z"Xu, Yuesheng"https://zbmath.org/authors/?q=ai:xu.yuesheng.1|xu.yueshengSummary: We consider a minimization problem whose objective function is the sum of a fidelity term, not necessarily convex, and a regularization term defined by a positive regularization parameter \(\lambda\) multiple of the \(\ell_0\) norm composed with a linear transform. This problem has wide applications in compressed sensing, sparse machine learning and image reconstruction. The goal of this paper is to understand what choices of the regularization parameter can dictate the level of sparsity under the transform for a global minimizer of the resulting regularized objective function. This is a critical issue but it has been left unaddressed. We address it from a geometric viewpoint with which the sparsity partition of the image space of the transform is introduced. Choices of the regularization parameter are specified to ensure that a global minimizer of the corresponding regularized objective function achieves a prescribed level of sparsity under the transform. Results are obtained for the spacial sparsity case in which the transform is the identity map, a case that covers several applications of practical importance, including machine learning, image/signal processing and medical image reconstruction.Global convergence of a class new smooth penalty algorithm for constrained optimization problemhttps://zbmath.org/1522.901472023-12-07T16:00:11.105023Z"Zhao, Wenling"https://zbmath.org/authors/?q=ai:zhao.wenling"Wang, Ruyu"https://zbmath.org/authors/?q=ai:wang.ruyu"Song, Daojin"https://zbmath.org/authors/?q=ai:song.daojinSummary: In this paper, a class of smooth penalty functions is proposed for constrained optimization problem. It is put forward based on \(L_p\), a smooth function of a class of exact penalty function \(\ell_p\) (\(p \in (0,1]\)). Based on the class of penalty functions, a penalty algorithm is presented. Under the very weak condition, a perturbation theorem is set up. The global convergence of the algorithm is derived. This result generalizes some existing conclusions. Finally, numerical experiments on two examples demonstrate the effectiveness and efficiency of our algorithm.A tight 4/3 approximation for capacitated vehicle routing in treeshttps://zbmath.org/1522.901482023-12-07T16:00:11.105023Z"Becker, Amariah"https://zbmath.org/authors/?q=ai:becker.amariahSummary: Given a set of clients with demands, the Capacitated Vehicle Routing problem is to find a set of tours that collectively cover all client demand, such that the capacity of each vehicle is not exceeded and such that the sum of the tour lengths is minimized. In this paper, we provide a 4/3-approximation algorithm for Capacitated Vehicle Routing on trees, improving over the previous best-known approximation ratio of \((\sqrt{41}-1)/4\) by \textit{T. Asano} et al. [STOC 1997, 275--283 (1999; Zbl 0962.68128)], while using the same lower bound. Asano et al. [loc. cit.] show that there exist instances whose optimal cost is 4/3 times this lower bound. Notably, our 4/3 approximation ratio is therefore tight for this lower bound, achieving the best-possible performance.
For the entire collection see [Zbl 1393.68012].On the competitiveness of memoryless strategies for the \(k\)-Canadian traveller problemhttps://zbmath.org/1522.901492023-12-07T16:00:11.105023Z"Bergé, Pierre"https://zbmath.org/authors/?q=ai:berge.pierre"Hemery, Julien"https://zbmath.org/authors/?q=ai:hemery.julien"Rimmel, Arpad"https://zbmath.org/authors/?q=ai:rimmel.arpad"Tomasik, Joanna"https://zbmath.org/authors/?q=ai:tomasik.joannaSummary: The \(k\)-Canadian Traveller Problem \((k\)-CTP), proven PSPACE-complete by Papadimitriou and Yannakakis, is a generalization of the Shortest Path Problem which admits blocked edges. Its objective is to determine the strategy that makes the traveller traverse graph \(G\) between two given nodes \(s\) and \(t\) with the minimal distance, knowing that at most \(k\) edges are blocked. The traveller discovers that an edge is blocked when arriving at one of its endpoints.
We study the competitiveness of randomized memoryless strategies to solve the \(k\)-CTP. Memoryless strategies are attractive in practice as a decision made by the strategy for a traveller in node \(v\) of \(G\) does not depend on his anterior moves. We establish that the competitive ratio of any randomized memoryless strategy cannot be better than \(2k + O(1) \). This means that randomized memoryless strategies are asymptotically as competitive as deterministic strategies which achieve a ratio \(2k+1\) at best.
For the entire collection see [Zbl 1407.68037].The approximation ratio of the \(k\)-Opt heuristic for the Euclidean traveling salesman problemhttps://zbmath.org/1522.901502023-12-07T16:00:11.105023Z"Brodowsky, Ulrich A."https://zbmath.org/authors/?q=ai:brodowsky.ulrich-a"Hougardy, Stefan"https://zbmath.org/authors/?q=ai:hougardy.stefan"Zhong, Xianghui"https://zbmath.org/authors/?q=ai:zhong.xianghuiSummary: The \(k\)-Opt heuristic is a simple improvement heuristic for the traveling salesman problem. It starts with an arbitrary tour and then repeatedly replaces \(k\) edges of the tour by \(k\) other edges, as long as this yields a shorter tour. We will prove that for the 2-dimensional Euclidean traveling salesman problem with \(n\) cities the approximation ratio of the \(k\)-Opt heuristic is \(\Theta(\log n/\log\log n)\). This improves the upper bound of \(O(\log n)\) given by \textit{B. Chandra} et al. in [SIAM J. Comput. 28, No. 6, 1998--2029 (1999; Zbl 0936.68052)] and provides for the first time a nontrivial lower bound for the case \(k\geq 3\). Our results not only hold for the Euclidean norm but extend to arbitrary \(p\)-norms with \(1\leq p<\infty\).Star routing: between vehicle routing and vertex coverhttps://zbmath.org/1522.901512023-12-07T16:00:11.105023Z"Delle Donne, Diego"https://zbmath.org/authors/?q=ai:donne.diego-delle"Tagliavini, Guido"https://zbmath.org/authors/?q=ai:tagliavini.guidoSummary: We consider an optimization problem posed by an actual newspaper company, which consists of computing a minimum length route for a delivery truck, such that the driver only stops at street crossings, each time delivering copies to all customers adjacent to the crossing. This can be modeled as an abstract problem that takes an unweighted simple graph \(G = (V, E)\) and a subset of edges \(X\) and asks for a shortest cycle, not necessarily simple, such that every edge of \(X\) has an endpoint in the cycle.
We show that the decision version of the problem is strongly NP-complete, even if \(G\) is a grid graph. Regarding approximate solutions, we show that the general case of the problem is APX-hard, and thus no PTAS is possible unless \(P = NP \). Despite the hardness of approximation, we show that given any \(\alpha \)-approximation algorithm for metric TSP, we can build a \(3\alpha \)-approximation algorithm for our optimization problem, yielding a concrete 9 / 2-approximation algorithm.
The grid case is of particular importance, because it models a city map or some part of it. A usual scenario is having some neighborhood full of customers, which translates as an instance of the abstract problem where almost every edge of \(G\) is in \(X\). We model this property as \(|E - X| = o(|E|)\), and for these instances we give a \((3/2 + \varepsilon )\)-approximation algorithm, for any \(\varepsilon > 0\), provided that the grid is sufficiently big.
For the entire collection see [Zbl 1407.68037].Critical node/edge detection problems on treeshttps://zbmath.org/1522.901522023-12-07T16:00:11.105023Z"Di Summa, Marco"https://zbmath.org/authors/?q=ai:di-summa.marco"Faruk, Syed Md Omar"https://zbmath.org/authors/?q=ai:faruk.syed-md-omarSummary: We consider the problem of removing a limited subset of nodes and/or edges from a graph in order to minimize the so-called pairwise connectivity of the residual graph, which is defined as the total cost of the pairs of nodes still connected by a path. This is a well-studied version of a family of problems known as critical node or edge detection problems. However, while most of the literature focuses on deleting nodes or edges separately, we allow the simultaneous removal of nodes and edges. We consider both the case in which the nodes and edges removed must satisfy a joint weight limit, and the case in which two separate weight limits are given for nodes and edges. We study the complexity of several problems of this type when the given graph is a tree, providing NP-hardness results or polynomial-time algorithms for the different cases that we analyze.Parallel batching with multi-size jobs and incompatible job familieshttps://zbmath.org/1522.901532023-12-07T16:00:11.105023Z"Druetto, Alessandro"https://zbmath.org/authors/?q=ai:druetto.alessandro"Pastore, Erica"https://zbmath.org/authors/?q=ai:pastore.erica"Rener, Elena"https://zbmath.org/authors/?q=ai:rener.elenaSummary: Parallel batch scheduling has many applications in the industrial sector, like in material and chemical treatments, mold manufacturing and so on. The number of jobs that can be processed on a machine mostly depends on the shape and size of the jobs and of the machine. This work investigates the problem of batching jobs with multiple sizes and multiple incompatible families. A flow formulation of the problem is exploited to solve it through two column generation-based heuristics. First, the column generation finds the optimal solution of the continuous relaxation, then two heuristics are proposed to move from the continuous to the integer solution of the problem: one is based on the price-and-branch heuristic, the other on a variable rounding procedure. Experiments with several combinations of parameters are provided to show the impact of the number of sizes and families on computation times and quality of solutions.Approximation algorithms for the load-balanced capacitated vehicle routing problemhttps://zbmath.org/1522.901542023-12-07T16:00:11.105023Z"Fallah, Haniyeh"https://zbmath.org/authors/?q=ai:fallah.haniyeh"Didehvar, Farzad"https://zbmath.org/authors/?q=ai:didehvar.farzad"Rahmati, Farhad"https://zbmath.org/authors/?q=ai:rahmati.farhadSummary: We study the load-balanced capacitated vehicle routing problem (LBCVRP): the problem is to design a collection of tours for a fixed fleet of vehicles with capacity \(Q\) to distribute a supply from a single depot between a number of predefined clients, in a way that the total traveling cost is a minimum, and the vehicle loads are balanced. The unbalanced loads cause the decrease of distribution quality especially in business environments and flexibility in the logistics activities. The problem being NP-hard, we propose two approximation algorithms. When the demands are equal, we present a \(((1-\frac{1}{Q})\rho +\frac{3}{2})\)-approximation algorithm that finds balanced loads. Here, \(\rho\) is the approximation ratio for the known metric traveling salesman problem (TSP). This result leads to a \(2.5-\frac{1}{Q}\) approximation ratio for the tree metrics since an optimal solution can be found for the TSP on a tree. We present an improved 2-approximation algorithm. When the demands are unequal, we focus on obtaining approximate solutions since finding balanced loads is NP-complete. We propose an algorithm that provides a 4-approximation for the balance of the loads. We assume a second approach to get around the difficulties of the feasibility. In this approach, we redefine and convert the problem into a multi-objective problem. The algorithm we propose has a 4 factor of approximation.Integer knapsack problems with profit functions of the same value rangehttps://zbmath.org/1522.901552023-12-07T16:00:11.105023Z"Gurevsky, Evgeny"https://zbmath.org/authors/?q=ai:gurevsky.evgeny-e"Kopelevich, Dmitry"https://zbmath.org/authors/?q=ai:kopelevich.dmitry-i"Kovalev, Sergey"https://zbmath.org/authors/?q=ai:kovalev.sergei-protasovich"Kovalyov, Mikhail Y."https://zbmath.org/authors/?q=ai:kovalyov.mikhail-yakovlevichSummary: Integer knapsack problems with profit functions of the same value range are studied. Linear time algorithms are presented for the case of convex non-decreasing profit functions, and an NP-hardness proof and a fully polynomial-time approximation scheme are provided for the case of arbitrary non-negative non-decreasing profit functions. Fast solution procedures are also devised for the bottleneck counterparts of these problems. Computational complexity of the case with concave profit functions remains open.A binary search double greedy algorithm for non-monotone DR-submodular maximizationhttps://zbmath.org/1522.901562023-12-07T16:00:11.105023Z"Gu, Shuyang"https://zbmath.org/authors/?q=ai:gu.shuyang"Gao, Chuangen"https://zbmath.org/authors/?q=ai:gao.chuangen"Wu, Weili"https://zbmath.org/authors/?q=ai:wu.weiliSummary: In this paper, we study the non-monotone DR-submodular function maximization over integer lattice. Functions over integer lattice have been defined submodular property that is similar to submodularity of set functions. DR-submodular is a further extended submodular concept for functions over the integer lattice, which captures the diminishing return property. Such functions finds many applications in machine learning, social networks, wireless networks, etc. The techniques for submodular set function maximization can be applied to DR-submodular function maximization, e.g., the double greedy algorithm has a \(1/2\)-approximation ratio, whose running time is \(O(nB)\), where \(n\) is the size of the ground set, \(B\) is the integer bound of a coordinate. In our study, we design a \(1/2\)-approximate binary search double greedy algorithm, and we prove that its time complexity is \(O(n\log B)\), which significantly improves the running time.
For the entire collection see [Zbl 1514.68006].Heuristics for the score-constrained strip-packing problemhttps://zbmath.org/1522.901572023-12-07T16:00:11.105023Z"Hawa, Asyl L."https://zbmath.org/authors/?q=ai:hawa.asyl-l"Lewis, Rhyd"https://zbmath.org/authors/?q=ai:lewis.rhyd"Thompson, Jonathan M."https://zbmath.org/authors/?q=ai:thompson.jonathan-mSummary: This paper investigates the Score-Constrained Strip-Packing Problem (SCSPP), a combinatorial optimisation problem that generalises the one-dimensional bin-packing problem. In the construction of cardboard boxes, rectangular items are packed onto strips to be scored by knives prior to being folded. The order and orientation of the items on the strips determine whether the knives are able to score the items correctly. Initially, we detail an exact polynomial-time algorithm for finding a feasible alignment of items on a single strip. We then integrate this algorithm with a packing heuristic to address the multi-strip problem and compare with two other greedy heuristics, discussing the circumstances in which each method is superior.
For the entire collection see [Zbl 1407.68037].Cyclic products and optimal traps in cyclic birth and death chainshttps://zbmath.org/1522.901582023-12-07T16:00:11.105023Z"Holmes, Mark"https://zbmath.org/authors/?q=ai:holmes.mark-d|holmes.mark-h|holmes.mark-j|holmes.mark-p"Holroyd, Alexander E."https://zbmath.org/authors/?q=ai:holroyd.alexander-e"Ramírez, Alejandro"https://zbmath.org/authors/?q=ai:ramirez.alejandro|ramirez.alejandro-fSummary: A birth-death chain is a discrete-time Markov chain on the integers whose transition probabilities \(p_{i,j}\) are non-zero if and only if \(|i-j|=1\). We consider birth-death chains whose birth probabilities \(p_{i,i+1}\) form a periodic sequence, so that \(p_{i,i+1}=p_{i \mod m}\) for some \(m\) and \(p_0,\ldots,p_{m-1}\). The trajectory \((X_n)_{n=0,1,\ldots}\) of such a chain satisfies a strong law of large numbers and a central limit theorem. We study the effect of reordering the probabilities \(p_0,\ldots,p_{m-1}\) on the velocity \(v=\lim_{n\to\infty} X_n/n\). The sign of \(v\) is not affected by reordering, but its magnitude in general is. We show that for Lebesgue almost every choice of \((p_0,\ldots,p_{m-1})\), exactly \((m-1)!/2\) distinct speeds can be obtained by reordering. We make an explicit conjecture of the ordering that minimises the speed, and prove it for all \(m\leqslant 7\). This conjecture is implied by a purely combinatorial conjecture that we think is of independent interest.Improved approximation algorithm for the asymmetric prize-collecting TSPhttps://zbmath.org/1522.901592023-12-07T16:00:11.105023Z"Hou, Bo"https://zbmath.org/authors/?q=ai:hou.bo"Pang, Zhenzhen"https://zbmath.org/authors/?q=ai:pang.zhenzhen"Gao, Suogang"https://zbmath.org/authors/?q=ai:gao.suogang"Liu, Wen"https://zbmath.org/authors/?q=ai:liu.wen.3Summary: We present a \(\frac{4\lceil \log (n)\rceil}{0.698\lceil \log (n)\rceil +1.302}\)-approximation algorithm for the asymmetric prize-collecting traveling salesman problem. This is obtained by combining a randomized variant of a rounding algorithm of \textit{V. H. Nguyen} and \textit{T. T. T. Nguyen} [Int. J. Math. Oper. Res. 4, No. 3, 294--301 (2012; Zbl 1269.90094)] and a primal-dual algorithm of \textit{V. H. Nguyen} [J. Comb. Optim. 25, No. 2, 265--278 (2013; Zbl 1268.90070)].
For the entire collection see [Zbl 1514.68006].Finding maximum edge-disjoint paths between multiple terminalshttps://zbmath.org/1522.901602023-12-07T16:00:11.105023Z"Iwata, Satoru"https://zbmath.org/authors/?q=ai:iwata.satoru"Yokoi, Yu"https://zbmath.org/authors/?q=ai:yokoi.yuSummary: Let \(G=(V,E)\) be a multigraph with a set \(T\subseteq V\) of terminals. A path in \(G\) is called a \(T\)-path if its ends are distinct vertices in \(T\) and no internal vertices belong to \(T\). In [Arch. Math. 30, 325--336 (1978; Zbl 0354.05042) and ``On the maximum number of intersection-free \(H\)-paths'' (German), ibid. 31, 387--402 (1978; \url{doi:10.1007/BF01226465})], \textit{W. Mader} showed a characterization of the maximum number of edge-disjoint \(T\)-paths. In this paper, we provide a combinatorial, deterministic algorithm for finding the maximum number of edge-disjoint \(T\)-paths. The algorithm adopts an augmenting path approach. More specifically, we utilize a new concept of short augmenting walks in auxiliary labeled graphs to capture a possible augmentation of the number of edge-disjoint \(T\)-paths. To design a search procedure for a short augmenting walk, we introduce blossoms analogously to the matching algorithm of \textit{J. Edmonds} [Can. J. Math. 17, 449--467 (1965; Zbl 0132.20903)]. When the search procedure terminates without finding a short augmenting walk, the algorithm provides a certificate for the optimality of the current edge-disjoint \(T\)-paths. From this certificate, one can obtain the Edmonds-Gallai type decomposition introduced by \textit{A. Sebő} and \textit{L. Szegő} [Lect. Notes Comput. Sci. 3064, 256--270 (2004; Zbl 1093.05512)]. The algorithm runs in \(O(|E|^2)\) time, which is much faster than the best known deterministic algorithm based on a reduction to linear matroid parity. We also present a strongly polynomial algorithm for the maximum integer free multiflow problem, which asks for a nonnegative integer combination of \(T\)-paths maximizing the sum of the coefficients subject to capacity constraints on the edges.Problem-specific branch-and-bound algorithms for the precedence constrained generalized traveling salesman problemhttps://zbmath.org/1522.901612023-12-07T16:00:11.105023Z"Khachay, Michael"https://zbmath.org/authors/?q=ai:khachay.michael"Ukolov, Stanislav"https://zbmath.org/authors/?q=ai:ukolov.stanislav"Petunin, Alexander"https://zbmath.org/authors/?q=ai:petunin.aleksandr-aleksandrovichSummary: The Generalized Traveling Salesman Problem (GTSP) is a well-known combinatorial optimization problem having numerous valuable practical applications in operations research. In the Precedence Constrained GTSP (PCGTSP), any feasible tour is restricted to visit all the clusters according to some given partial order. Unlike the common setting of the GTSP, the PCGTSP appears still weakly studied in terms of algorithmic design and implementation. To the best of our knowledge, all the known algorithmic results for this problem can be exhausted by Salmans's general branching framework, a few MILP models, and the PCGLNS meta-heuristic proposed by the authors recently. In this paper, we present the first problem-specific branch-and-bound algorithm designed with an extension of Salman's approach and exploiting PCGLNS as a powerful primal heuristic. Using the public PCGTSPLIB testbench, we evaluate the performance of the proposed algorithm against the classic Held-Karp dynamic programming scheme with branch-and-bound node fathoming strategy and Gurobi state-of-the-art solver armed by our recently proposed MILP model and PCGLNS-based warm start.
For the entire collection see [Zbl 1508.90001].Energy-constrained geometric coverage problemhttps://zbmath.org/1522.901622023-12-07T16:00:11.105023Z"Lan, Huan"https://zbmath.org/authors/?q=ai:lan.huanSummary: Wireless sensor networks have many applications in real life. We are given \(m\) sensors and \(n\) users on the plane. The coverage of each sensor \(s\) is a disc area, whose radius \(r(s)\) and energy \(p(s)\) satisfy that \(p(s)= r(s)^\alpha\), where \(\alpha \ge 1\) is the attenuation factor. In this paper, we study the energy-constrained geometric coverage problem, which is to find an energy allocation scheme such that the total energy does not exceed a given bound P, and the total profit of the covered points is maximized. We propose a greedy algorithm whose approximation ratio is \(1-\frac{1}{\sqrt{e}}\).
For the entire collection see [Zbl 1514.68006].Effect of crowd composition on the wisdom of artificial crowds metaheuristichttps://zbmath.org/1522.901632023-12-07T16:00:11.105023Z"Lowrance, Christopher J."https://zbmath.org/authors/?q=ai:lowrance.christopher-j"Larkin, Dominic M."https://zbmath.org/authors/?q=ai:larkin.dominic-m"Yim, Sang M."https://zbmath.org/authors/?q=ai:yim.sang-mSummary: This paper investigates the impact that task difficulty and crowd composition have on the success of the \textit{Wisdom of Artificial Crowds} metaheuristic. The metaheuristic, which is inspired by the \textit{wisdom of crowds} phenomenon, combines the intelligence from a group of optimization searches to form a new solution. Unfortunately, the aggregate formed by the metaheuristic is not always better than the best individual solution within the crowd, and little is known about the variables which maximize the metaheuristic's success. Our study offers new insights into the influential factors of artificial crowds and the collective intelligence of multiple optimization searches performed on the same problem. The results show that favoring the opinions of experts (i.e., the better searches) improves the chances of the metaheuristic succeeding by more than 15\% when compared to the traditional means of equal weighting. Furthermore, weighting expertise was found to require smaller crowd sizes for the metaheuristic to reach its peak chances of success. Finally, crowd size was discovered to be a critical factor, especially as problem complexity grows or average crowd expertise declines. However, crowd size matters only up to a point, after which the probability of success plateaus.
For the entire collection see [Zbl 1407.68037].Primal dual algorithm for partial set multi-coverhttps://zbmath.org/1522.901642023-12-07T16:00:11.105023Z"Ran, Yingli"https://zbmath.org/authors/?q=ai:ran.yingli"Shi, Yishuo"https://zbmath.org/authors/?q=ai:shi.yishuo"Zhang, Zhao"https://zbmath.org/authors/?q=ai:zhang.zhaoSummary: In a minimum partial set multi-cover problem (MinPSMC), given an element set \(E\), a collection of subsets \(\mathcal{S}\subseteq 2^E\), a cost \(w_S\) on each set \(S\in \mathcal{S}\), a covering requirement \(r_e\) for each element \(e\in E\), and an integer \(k\), the goal is to find a sub-collection \(\mathcal{F}\subseteq \mathcal{S}\) to fully cover at least \(k\) elements such that the cost of \(\mathcal{F}\) is as small as possible, where element \(e\) is fully covered by \(\mathcal{F}\) if it belongs to at least \(r_e\) sets of \(\mathcal{F}\). On the application side, the problem has its background in the seed selection problem in a social network. On the theoretical side, it is a natural combination of the minimum partial (single) set cover problem (MinPSC) and the minimum set multi-cover problem (MinSMC). Although both MinPSC and MinSMC admit good approximations whose performance ratios match those lower bounds for the classic set cover problem, previous studies show that theoretical study on MinPSMC is quite challenging. In this paper, we prove that MinPSMC cannot be approximated within factor \(O(n^\frac{1}{2(\log \log n)^c})\) under the ETH assumption. A primal dual algorithm for MinPSMC is presented with a guaranteed performance ratio \(O(\sqrt{n})\) when \(r_{\max }\) and \(f\) are constants, where \(r_{\max } =\max _{e\in E} r_e\) is the maximum covering requirement and \(f\) is the maximum frequency of elements (that is the maximum number of sets containing a common element). We also improve the ratio for a restricted version of MinPSMC which possesses a graph-type structure.
For the entire collection see [Zbl 1407.68037].Round-robin scheduling with regard to rest differenceshttps://zbmath.org/1522.901652023-12-07T16:00:11.105023Z"Tuffaha, Tasbih"https://zbmath.org/authors/?q=ai:tuffaha.tasbih"Çavdaroğlu, Burak"https://zbmath.org/authors/?q=ai:cavdaroglu.burak"Atan, Tankut"https://zbmath.org/authors/?q=ai:atan.tankut-sSummary: Fairness is a key consideration in designing sports schedules. When two teams play against each other, it is only fair to let them rest the same amount of time before their game. In this study, we aim to reduce, if not eliminate, the difference between the rest durations of opposing teams in each game of a round-robin tournament. The rest difference problem proposed here constructs a timetable that determines both the round and the matchday of each game such that the total rest difference throughout the tournament is minimized. We provide a mixed-integer programming formulation and a matheuristic algorithm that tackle the rest difference problem. Moreover, we develop a polynomial-time exact algorithm for some special cases of the problem. This algorithm finds optimal schedules with zero total rest difference when the number of teams is a positive-integer power of 2 and the number of games in each day is even. Some theoretical results regarding tournaments with one-game matchdays are also presented.Lexicographic optimization for the multi-container loading problem with open dimensions for a shoe manufacturerhttps://zbmath.org/1522.901662023-12-07T16:00:11.105023Z"Vieira, Manuel V. C."https://zbmath.org/authors/?q=ai:vieira.manuel-v-c"Carvalho, Margarida"https://zbmath.org/authors/?q=ai:carvalho.margaridaSummary: Motivated by a real-world application, we present a multi-container loading problem with 3-open dimensions. We formulate it as a biobjective mixed-integer nonlinear program with lexicographic objectives in order to reflect the decision maker's optimization priorities. The first objective is to minimize the number of containers, while the second objective is to minimize the volume of those containers. Besides showing the NP-hardness of this sequential optimization problem, we provide bounds for it which are used in the three proposed algorithms, as well as, on their evaluation when a certificate of optimality is not available. The first is an exact parametric-based approach to tackle the lexicographic optimization through the second objective of the problem. Nevertheless, given that the parametric programs correspond to large nonlinear mixed-integer optimizations, we present a heuristic that is entirely mathematical-programming based. The third algorithm enhances the solution quality of the heuristic. These algorithms are specifically tailored for the real-world application. The effectiveness and efficiency of the devised heuristics is demonstrated with numerical experiments.Improving quantum computation by optimized qubit routinghttps://zbmath.org/1522.901672023-12-07T16:00:11.105023Z"Wagner, Friedrich"https://zbmath.org/authors/?q=ai:wagner.friedrich-g"Bärmann, Andreas"https://zbmath.org/authors/?q=ai:barmann.andreas"Liers, Frauke"https://zbmath.org/authors/?q=ai:liers.frauke"Weissenbäck, Markus"https://zbmath.org/authors/?q=ai:weissenback.markusSummary: In this work we propose a high-quality decomposition approach for qubit routing by swap insertion. This optimization problem arises in the context of compiling quantum algorithms formulated in the circuit model of computation onto specific quantum hardware. Our approach decomposes the routing problem into an allocation subproblem and a set of token swapping problems. This allows us to tackle the allocation part and the token swapping part separately. Extracting the allocation part from the qubit routing model of \textit{G. Nannicini} et al. [``Optimal qubit assignment and routing via integer programming'', Preprint, \url{arXiv:2106.06446}], we formulate the allocation subproblem as a binary linear program. Herein, we employ a cost function that is a lower bound on the overall routing problem objective. We strengthen the linear relaxation by novel valid inequalities. For the token swapping part we develop an exact branch-and-bound algorithm. In this context, we improve upon known lower bounds on the token swapping problem. Furthermore, we enhance an existing approximation algorithm which runs much faster than the exact approach and typically is able to determine solutions close to the optimum. We present numerical results for the fully integrated allocation and token swapping problem. Obtained solutions may not be globally optimal due to the decomposition and the usage of an approximation algorithm. However, the solutions are obtained fast and are typically close to optimal. In addition, there is a significant reduction in the number of artificial gates and output circuit depth when compared to various state-of-the-art heuristics. Reducing these figures is crucial for minimizing noise when running quantum algorithms on near-term hardware. As a consequence, using the novel decomposition approach leads to compiled algorithms with improved quality. Indeed, when compiled with the novel routing procedure and executed on real hardware, our experimental results for quantum approximate optimization algorithms show an significant increase in solution quality in comparison to standard routing methods.Zero-norm regularized problems: equivalent surrogates, proximal MM method and statistical error boundhttps://zbmath.org/1522.901682023-12-07T16:00:11.105023Z"Zhang, Dongdong"https://zbmath.org/authors/?q=ai:zhang.dongdong"Pan, Shaohua"https://zbmath.org/authors/?q=ai:pan.shaohua"Bi, Shujun"https://zbmath.org/authors/?q=ai:bi.shujun"Sun, Defeng"https://zbmath.org/authors/?q=ai:sun.defengSummary: For the zero-norm regularized problem, we verify that the penalty problem of its equivalent MPEC reformulation is a global exact penalty, which implies a family of equivalent surrogates. For a subfamily of these surrogates, the critical point set is demonstrated to coincide with the \(d\)-directional stationary point set and when a critical point has no too small nonzero component, it is a strongly local optimal solution of the surrogate problem and the zero-norm regularized problem. We also develop a proximal majorization-minimization (MM) method for solving the DC (difference of convex functions) surrogates, and provide its global and linear convergence analysis. For the limit of the generated sequence, the statistical error bound is established under a mild condition, which implies its good quality from a statistical respective. Numerical comparisons with ADMM for solving the DC surrogate and APG for solving its partially smoothed form indicate that our proximal MM method armed with an inexact dual PPA plus the semismooth Newton method (PMMSN for short) is remarkably superior to ADMM and APG in terms of the quality of solutions and the CPU time.An approximation algorithm for the clustered path travelling salesman problemhttps://zbmath.org/1522.901692023-12-07T16:00:11.105023Z"Zhang, Jiaxuan"https://zbmath.org/authors/?q=ai:zhang.jiaxuan"Gao, Suogang"https://zbmath.org/authors/?q=ai:gao.suogang"Hou, Bo"https://zbmath.org/authors/?q=ai:hou.bo"Liu, Wen"https://zbmath.org/authors/?q=ai:liu.wen.3Summary: In this paper, we consider the clustered path travelling salesman problem. In this problem, we are given a complete graph \(G=(V,E)\) with edge weight satisfying the triangle inequality. In addition, the vertex set \(V\) is partitioned into clusters \(V_1,\cdots ,V_k\). The objective of the problem is to find a minimum Hamiltonian path in \(G\), and in the path all vertices of each cluster are visited consecutively. We provide a polynomial-time approximation algorithm for the problem.
For the entire collection see [Zbl 1514.68006].A combinatorial approximation algorithm for \(k\)-level facility location problem with submodular penaltieshttps://zbmath.org/1522.901702023-12-07T16:00:11.105023Z"Zhang, Li"https://zbmath.org/authors/?q=ai:zhang.li.1|zhang.li.10|zhang.li.16|zhang.li.61|zhang.li.11|zhang.li.35|zhang.li.28|zhang.li.6|zhang.li.40|zhang.li.27|zhang.li.9|zhang.li.17|zhang.li.15|zhang.li.7|zhang.li.3|zhang.li.13|zhang.li.14|zhang.li.8|zhang.li.5|zhang.li.20"Yuan, Jing"https://zbmath.org/authors/?q=ai:yuan.jing"Xu, Zhizhen"https://zbmath.org/authors/?q=ai:xu.zhizhen"Li, Qiaoliang"https://zbmath.org/authors/?q=ai:li.qiaoliangSummary: We present an improved approximation algorithm for \(k\)-level facility location problem with submodular penalties, the new approximation ratio is 2.9444 for any constant \(k\), which improves the current best approximation ratio 3.314. The central ideas in our results are as follows: first, we restructure the problem as an uncapacitated facility location problem, then we use the primal-dual scheme with greedy augmentation. The key technique of our result is that we change the way of last opening facility set in primal-dual approximation algorithm to get much more tight result for \(k\)-level facility location problem with submodular penalties.Simultaneous eating algorithm and greedy algorithm in assignment problemshttps://zbmath.org/1522.901712023-12-07T16:00:11.105023Z"Zhan, Ping"https://zbmath.org/authors/?q=ai:zhan.pingSummary: The simultaneous eating algorithm (SEA) and probabilistic serial (PS) mechanism are well known for allocating a set of divisible or indivisible goods to agents with ordinal preferences. The PS mechanism is SEA at the same eating speed. The prominent property of SEA is ordinal efficiency. Recently, we extended the PS mechanism (EPS) from a fixed quota of each good to a variable varying in a polytope constrained by submodular functions. In this article, we further generalized some results on SEA. After formalizing the extended ESA (ESEA), we show that it can be characterized by ordinal efficiency. We established a stronger summation optimization than the Pareto type of ordinal efficiency by an introduced weight vector. The weights in the summation optimization coincide with agents' preferences at the acyclic positive values of an allocation. Hence, the order of goods selected to eat in ESEA is exactly the one chosen in the execution of the well-known greedy algorithm.On some properties of conjugate relation and subdifferentials in set optimization problemhttps://zbmath.org/1522.901722023-12-07T16:00:11.105023Z"Araya, Yousuke"https://zbmath.org/authors/?q=ai:araya.yousukeSummary: We first give new definitions of set-valued conjugate relation based on comparison of sets introduced by \textit{J. Jahn} and \textit{T. X. D. Ha} in [J. Optim. Theory Appl. 148, No. 2, 209--236 (2011; Zbl 1226.90092)]. Then we give some types of weak duality theorems. Next, by using nonlinear scalarizing technique for set, we present a strong duality theorem. We also give some continuity properties of conjugate relation for set-valued map. Lastly, we give some definitions of subdifferentials for set-valued map and investigate its properties.
For the entire collection see [Zbl 1480.00062].A new scheme for approximating the weakly efficient solution set of vector rational optimization problemshttps://zbmath.org/1522.901732023-12-07T16:00:11.105023Z"Guo, Feng"https://zbmath.org/authors/?q=ai:guo.feng"Jiao, Liguo"https://zbmath.org/authors/?q=ai:jiao.liguoSummary: In this paper, we provide a new scheme for approximating the weakly efficient solution set for a class of vector optimization problems with rational objectives over a feasible set defined by finitely many polynomial inequalities. More precisely, we present a procedure to obtain a sequence of explicit approximations of the weakly efficient solution set of the problem in question. Each approximation is the intersection of the sublevel set of a single polynomial and the feasible set. To this end, we make use of the achievement function associated with the considered problem and construct polynomial approximations of it over the feasible set from above. Remarkably, the construction can be converted to semidefinite programming problems. Several nontrivial examples are designed to illustrate the proposed new scheme.Spectral conjugate gradient methods for vector optimization problemshttps://zbmath.org/1522.901742023-12-07T16:00:11.105023Z"He, Qing-Rui"https://zbmath.org/authors/?q=ai:he.qing-rui"Chen, Chun-Rong"https://zbmath.org/authors/?q=ai:chen.chunrong"Li, Sheng-Jie"https://zbmath.org/authors/?q=ai:li.shengjieSummary: In this work, we present an extension of the spectral conjugate gradient (SCG) methods for solving unconstrained vector optimization problems, with respect to the partial order induced by a pointed, closed and convex cone with a nonempty interior. We first study the direct extension version of the SCG methods and its global convergence without imposing an explicit restriction on parameters. It shows that the methods may lose their good scalar properties, like yielding descent directions, in the vector setting. By using a truncation technique, we then propose a modified self-adjusting SCG algorithm which is more suitable for various parameters. Global convergence of the new scheme covers the vector extensions of three different spectral parameters and the corresponding Perry, Andrei, and Dai-Kou conjugate parameters (SP, N, and JC schemes, respectively) without regular restarts and any convex assumption. Under inexact line searches, we prove that the sequences generated by the proposed methods find points that satisfy the first-order necessary condition for Pareto-optimality. Finally, numerical experiments illustrating the practical behavior of the methods are presented.On a class of nonsmooth fractional robust multi-objective optimization problems. II: Dualityhttps://zbmath.org/1522.901752023-12-07T16:00:11.105023Z"Hong, Zhe"https://zbmath.org/authors/?q=ai:hong.zhe"Bae, Kwan Deok"https://zbmath.org/authors/?q=ai:bae.kwan-deok"Jiao, Liguo"https://zbmath.org/authors/?q=ai:jiao.liguo"Kim, Do Sang"https://zbmath.org/authors/?q=ai:kim.do-sangSummary: In the previous paper [\textit{Z. Hong} et al., in: Proceedings of the 11th international conference on nonlinear analysis and convex analysis (NACA 2019) and the International conference on optimization: techniques and applications (ICOTA 2019). Part I. Yokohama: Yokohama Publishers. 101--118 (2019; Zbl 07716269)], the authors did some works on optimality conditions for a class of nonsmooth fractional robust multi-objective optimization problems. In this paper, we further study duality results for such a class of optimization problems. More precisely, we propose model-types of both non-fractional and fractional dual problems; then weak, strong, and converse-like duality relations are investigated, respectively.
For the entire collection see [Zbl 1480.00062].Slack-based generalized Tchebycheff norm scalarization approaches for solving multiobjective optimization problemshttps://zbmath.org/1522.901762023-12-07T16:00:11.105023Z"Hoseinpoor, N."https://zbmath.org/authors/?q=ai:hoseinpoor.narges"Ghaznavi, M."https://zbmath.org/authors/?q=ai:ghaznavi.mehrdad(no abstract)On nonsmooth minimax semi-infinite programming problems with mixed constraintshttps://zbmath.org/1522.901772023-12-07T16:00:11.105023Z"Jaisawal, Pushkar"https://zbmath.org/authors/?q=ai:jaisawal.pushkar"Singh, Harsh Narayan"https://zbmath.org/authors/?q=ai:singh.harsh-narayan"Laha, Vivek"https://zbmath.org/authors/?q=ai:laha.vivekSummary: We consider a nonsmooth minimax semi-infinite programming problem with mixed constraints (NMMSIP) and study for the first time in literature with the use of Michel-Penot (M-P) subdifferential. We establish the Karush-Kuhn-Tucker type necessary optimality conditions for a point to be an optimal solution of the NMMSIP in terms of the smallest convex valued M-P subdifferentials. We establish sufficient type optimality conditions for the NMMSIP under convexity and generalized convexity assumptions in terms of M-P subdifferentials. Further, we formulate Wolfe type dual model (NMMSIP-WD) for NMMSIP and establish weak, strong, converse, restricted converse and strict converse duality results between NMMSIP-WD and NMMSIP under convexity assumptions in terms of M-P subdifferentials. Lastly, we establish Karush-Kuhn-Tucker type necessary optimality conditions for a nonsmooth multiobjective semi-infinite programming problem with mixed constraints (NMOSIP) using M-P subdifferentials and sufficient type optimality conditions for NMOSIP under convexity and generalized convexity assumptions in terms of M-P subdifferentials with the help of above established Karush-Kuhn-Tucker type necessary and sufficient optimality conditions for NMMSIP.
For the entire collection see [Zbl 1480.00062].Necessary and sufficient optimality conditions and a new approach for solving the smooth multiobjective fractional continuous-time programming problemhttps://zbmath.org/1522.901782023-12-07T16:00:11.105023Z"Jović, Aleksandar"https://zbmath.org/authors/?q=ai:jovic.aleksandarSummary: The multiobjective fractional continuous-time programming problem with inequality constraints is considered. We investigate the optimality conditions for this problem under weaker assumptions than in [\textit{G. J. Zalmai}, Optimization 37, No. 1, 1--25 (1996; Zbl 0867.90100)]. Necessary optimality conditions are obtained under a suitable constraint qualification and a certain regularity condition without convexity/concavity assumptions. It is important to highlight that the assumptions of convexity/concavity on objective and constraint functions in [loc. cit.] are stronger than the assumptions in this paper. Here, there are no assumptions of convexity/concavity for deriving necessary optimality conditions. Also, the constraint qualifications and a certain regularity condition presented in this paper are much less restrictive and easier to verify than the constraint qualifications given in [loc. cit.]. This means that the necessary optimality conditions, set in this paper, are obtained under the weakest possible assumptions that are known to date. The already achieved results in the area of multiobjective fractional continuous-time programming are improved and more generalized in this paper. Also, we provide several examples to illustrate our results.A note on characterization of (weakly/properly/robust) efficient solutions in nonsmooth semi-infinite multiobjective optimization using convexificatorshttps://zbmath.org/1522.901792023-12-07T16:00:11.105023Z"Kabgani, Alireza"https://zbmath.org/authors/?q=ai:kabgani.alireza"Soleimani-damaneh, Majid"https://zbmath.org/authors/?q=ai:soleimani-damaneh.majidSummary: This note concerns with Theorem 4.1 in [\textit{A. Kabgani} and \textit{M. Soleimani-damaneh}, Optimization 67, No. 2, 217--235 (2018; Zbl 1421.90135)]. We present this theorem in a correct way and show that the error does not influence other results of the paper. Moreover, we provide some new corollaries for this theorem.Modification of some scalarization approaches for multiobjective optimizationhttps://zbmath.org/1522.901802023-12-07T16:00:11.105023Z"Khorasani, Vahid Amiri"https://zbmath.org/authors/?q=ai:khorasani.vahid-amiri"Khorram, Esmaile"https://zbmath.org/authors/?q=ai:khorram.esmaileSummary: In this paper, we propose revisions of two existing scalarization approaches, namely the feasible-value constraint and the weighted constraint. These methods do not easily provide results on proper efficient solutions of a general multiobjective optimization problem. By proposing some novel modifications for these methods, we derive some interesting results concerning proper efficient solutions. These scalarization approaches need no convexity assumption of the objective functions. We also demonstrate the efficiency of the proposed method using numerical experiments. In particular, a rocket injector design problem involving four objective functions illustrates the performance of the proposed method.On nonsmooth multiobjective semidefinite programs and vector variational inequalities using convexificatorshttps://zbmath.org/1522.901812023-12-07T16:00:11.105023Z"Laha, Vivek"https://zbmath.org/authors/?q=ai:laha.vivek"Kumar, Rahul"https://zbmath.org/authors/?q=ai:kumar.rahul"Pandey, Lalita"https://zbmath.org/authors/?q=ai:pandey.lalitaSummary: We deal with nonsmooth multiobjective semidefinite programs (NMSDPs) and vector variational inequalities (VVIs) in terms of convexificators. We extend the notions of strict minimizers and semi-strict minimizers of higher order for the NMSDPs under the assumptions of strong convexity and strict strong convexity of higher order. We characterize strong convexity and strict strong convexity of a real valued function over the space of all symmetric matrices by the higher order monotonicity and strict monotonicity of its convexificator. We formulate several Stampacchia and Minty type VVIs of higher order in terms of convexificators and use them to characterize strict minimizers and semi-strict minimizers of higher order for the NMSDPs.
For the entire collection see [Zbl 1480.00062].Existence of solutions of set quasi-optimization problems involving Minkowski differencehttps://zbmath.org/1522.901822023-12-07T16:00:11.105023Z"Le Anh Tuan"https://zbmath.org/authors/?q=ai:le-anh-tuan.Summary: This paper gives verifiable conditions for the existence of solutions of some set quasi-optimization problems involving Minkowski difference, where the objective maps are defined in complete metric spaces. Our proof method is not based on any scalarizing approach, and our existence results are written in terms of the given data of the problems. Several examples are provided. As applications of the main results of this paper, new results on the existence of robust solutions for uncertain multi-objective quasi-optimization problems with set-valued maps are formulated.Frank-Wolfe type theorems for polynomial vector optimization problemshttps://zbmath.org/1522.901832023-12-07T16:00:11.105023Z"Liu, Dan-Yang"https://zbmath.org/authors/?q=ai:liu.danyang"Huang, La"https://zbmath.org/authors/?q=ai:huang.la"Hu, Rong"https://zbmath.org/authors/?q=ai:hu.rongSummary: We study the solvability of a polynomial vector optimization problem under the weak section-boundedness from below condition. We give a characterization of the weak section-boundedness from below condition. Under the weak section-boundedness condition, we prove the existence of weakly Pareto efficient solutions for a convex polynomial vector optimization problem. For the non-convex case, we prove the existence of Pareto efficient solutions when the convenience, non-degeneracy, and weak section-boundedness conditions are satisfied.A new model to protect an important node against two threatening agentshttps://zbmath.org/1522.901842023-12-07T16:00:11.105023Z"Maleki, Z."https://zbmath.org/authors/?q=ai:maleki.zahra"Maleki, H. R."https://zbmath.org/authors/?q=ai:maleki.hamid-reza"Akbari, R."https://zbmath.org/authors/?q=ai:akbari.reza(no abstract)Generalized polarity and weakest constraint qualifications in multiobjective optimizationhttps://zbmath.org/1522.901852023-12-07T16:00:11.105023Z"Stein, Oliver"https://zbmath.org/authors/?q=ai:stein.oliver.1|stein.oliver.2|stein.oliver-t"Volk, Maximilian"https://zbmath.org/authors/?q=ai:volk.maximilianSummary: In Haeser and Ramos (J Optim Theory Appl, 187:469--487, 2020), a generalization of the normal cone from single objective to multiobjective optimization is introduced, along with a weakest constraint qualification such that any local weak Pareto optimal point is a weak Kuhn-Tucker point. We extend this approach to other generalizations of the normal cone and corresponding weakest constraint qualifications, such that local Pareto optimal points are weak Kuhn-Tucker points, local proper Pareto optimal points are weak and proper Kuhn-Tucker points, respectively, and strict local Pareto optimal points of order one are weak, proper and strong Kuhn-Tucker points, respectively. The constructions are based on an appropriate generalization of polarity to pairs of matrices and vectors.An accelerated proximal gradient method for multiobjective optimizationhttps://zbmath.org/1522.901862023-12-07T16:00:11.105023Z"Tanabe, Hiroki"https://zbmath.org/authors/?q=ai:tanabe.hiroki"Fukuda, Ellen H."https://zbmath.org/authors/?q=ai:fukuda.ellen-hidemi"Yamashita, Nobuo"https://zbmath.org/authors/?q=ai:yamashita.nobuoSummary: This paper presents an accelerated proximal gradient method for multiobjective optimization, in which each objective function is the sum of a continuously differentiable, convex function and a closed, proper, convex function. Extending first-order methods for multiobjective problems without scalarization has been widely studied, but providing accelerated methods with accurate proofs of convergence rates remains an open problem. Our proposed method is a multiobjective generalization of the accelerated proximal gradient method, also known as the Fast Iterative Shrinkage-Thresholding Algorithm, for scalar optimization. The key to this successful extension is solving a subproblem with terms exclusive to the multiobjective case. This approach allows us to demonstrate the global convergence rate of the proposed method \((O(1/k^2))\), using a merit function to measure the complexity. Furthermore, we present an efficient way to solve the subproblem via its dual representation, and we confirm the validity of the proposed method through some numerical experiments.Necessary optimality conditions for second-order local strict efficiency for constrained nonsmooth vector equilibrium problemshttps://zbmath.org/1522.901872023-12-07T16:00:11.105023Z"Van Su, Tran"https://zbmath.org/authors/?q=ai:su.tran-van"Hang, Dinh Dieu"https://zbmath.org/authors/?q=ai:hang.dinh-dieuSummary: This paper is concerned with primal and dual second-order optimality conditions for the second-order strict efficiency of nonsmooth vector equilibrium problem with set, cone and equality conditions. First, we propose some second-order constraint qualifications via the second-order tangent sets. Second, we establish necessary optimality conditions of order two in terms of second-order contingent derivatives and second-order Shi sets for a second-order strict local Pareto minima to such problem under suitable assumptions on the second-order constraint qualifications. An application of the result for the twice Fréchet differentiable functions for the second-order local strict efficiency of that problem is also presented. Some illustrative examples are also provided for our findings.Convergence of inexact steepest descent algorithm for multiobjective optimizations on Riemannian manifolds without curvature constraintshttps://zbmath.org/1522.901882023-12-07T16:00:11.105023Z"Wang, X. M."https://zbmath.org/authors/?q=ai:wang.xiangmei"Wang, J. H."https://zbmath.org/authors/?q=ai:wang.jinhua.1"Li, C."https://zbmath.org/authors/?q=ai:li.chong.2Summary: We study the issue of convergence for inexact steepest descent algorithm (employing general step sizes) for multiobjective optimizations on general Riemannian manifolds (without curvature constraints). Under the assumption of the local convexity/quasi-convexity, local/global convergence results are established. Furthermore, without the assumption of the local convexity/quasi-convexity, but under an error bound-like condition, local/global convergence results and convergence rate estimates are presented, which are new even in the linear space setting. Our results improve/extend the corresponding ones in [\textit{J. Wang} et al., SIAM J. Optim. 31, No. 1, 172--199 (2021; Zbl 1457.53022)] for scalar optimization problems on Riemannian manifolds to multiobjective ones. Finally, for the special case when the inexact steepest descent algorithm employing Armijo rule, our results improve/extend the corresponding ones in [\textit{O. P. Ferreira} et al., J. Optim. Theory Appl. 184, No. 2, 507--533 (2020; Zbl 1432.90137)] by relaxing curvature constraints.On approximate optimality conditions for robust multi-objective convex optimization problemshttps://zbmath.org/1522.901892023-12-07T16:00:11.105023Z"Wu, Pengcheng"https://zbmath.org/authors/?q=ai:wu.pengcheng"Jiao, Liguo"https://zbmath.org/authors/?q=ai:jiao.liguo"Zhou, Yuying"https://zbmath.org/authors/?q=ai:zhou.yuyingSummary: In this paper, we are interested in the study of approximate optimality conditions for weakly \(\epsilon\)-efficient solutions to robust multi-objective optimization problems ((RMOP) for short) in view of its associated \textit{minimax optimization problem} (MMOP). To this end, we first establish the relationship between a weakly \(\epsilon\)-efficient solution to the problem (RMOP) and an \(\alpha\)-solution to the problem (MMOP), where \(\epsilon = (\epsilon_1, \ldots, \epsilon_p) \in \mathbb{R}_+^p \setminus \{ 0\}\) and \(\alpha =\max_{j=1,\ldots, p} \{ \epsilon_j \}\). Then, we explore the representation of the so-called \(\beta\)-normal set (where \(\beta\geq 0\) is a given parameter) to a closed convex set at some reference point by two methods. At last, by employing the \(\alpha\)-subdifferential of the max-function and the obtained representation of the \(\beta\)-normal set, we establish an approximate necessary optimality condition for the problem (RMOP). Moreover, we also give an example to illustrate our results.Optimality conditions for multiobjective optimization problems via image space analysishttps://zbmath.org/1522.901902023-12-07T16:00:11.105023Z"Xu, Yingrang"https://zbmath.org/authors/?q=ai:xu.yingrang"Li, Shengjie"https://zbmath.org/authors/?q=ai:li.shengjieSummary: In this article, optimality conditions on (weak) efficient solutions in multiobjective optimization problems are investigated by using the image space analysis. A class of strong separation functions is constructed by oriented distance functions. Simultaneously, a generalized Lagrange function is introduced by the class of strong separation functions. Then, generalized Karush-Kuhn-Tucker (KKT for short) necessary optimality conditions are established without constraint qualifications or regularity conditions. Under the suitable assumptions, Lagrangian-type sufficient optimality conditions are also characterized. Moreover, the difference between strong separation and weak separation methods is explained.A bundle-like progressive hedging algorithmhttps://zbmath.org/1522.901912023-12-07T16:00:11.105023Z"Atenas, Felipe"https://zbmath.org/authors/?q=ai:atenas.felipe"Sagastizábal, Claudia"https://zbmath.org/authors/?q=ai:sagastizabal.claudia-aSummary: For convex multistage programming problems, we propose a variant for the Progressive Hedging algorithm inspired from bundle methods. Like in the original algorithm, iterates are generated by first solving separate problems for each scenario, and then performing a projective step to ensure non-anticipativity. An additional test checks the quality of the approximation, splitting iterates into two subsequences, akin to the dichotomy between bundle serious and null steps. The method is shown to converge in both cases, and the convergence rate is linear for the serious subsequence. Our bundle-like approach endows the Progressive Hedging algorithm with an implementable stopping test. Moreover, it is possible to vary the augmentation parameter along iterations without impairing convergence. Such enhancements with respect to the original Progressive Hedging algorithm are obtained at the expense of the solution of additional subproblems at each iteration, one per scenario.Second-order enhanced optimality conditions and constraint qualificationshttps://zbmath.org/1522.901922023-12-07T16:00:11.105023Z"Bai, Kuang"https://zbmath.org/authors/?q=ai:bai.kuang"Song, Yixia"https://zbmath.org/authors/?q=ai:song.yixia"Zhang, Jin"https://zbmath.org/authors/?q=ai:zhang.jin.2Summary: In this paper, we study second-order necessary optimality conditions for smooth nonlinear programming problems. Employing the second-order variational analysis and generalized differentiation, under the weak constant rank (WCR) condition, we derive an enhanced version of the classical weak second-order Fritz-John condition which contains some new information on multipliers. Based on this enhanced weak second-order Fritz-John condition, we introduce the weak second-order enhanced Karush-Kuhn-Tucker condition and propose some associated second-order constraint qualifications. Finally, using our new second-order constraint qualifications, we establish new sufficient conditions for the existence of a Hölder error bound condition.Accelerated smoothing hard thresholding algorithms for \(\ell_0\) regularized nonsmooth convex regression problemhttps://zbmath.org/1522.901932023-12-07T16:00:11.105023Z"Bian, Wei"https://zbmath.org/authors/?q=ai:bian.wei"Wu, Fan"https://zbmath.org/authors/?q=ai:wu.fanSummary: We study a class of constrained sparse optimization problems with cardinality penalty, where the feasible set is defined by box constraint, and the loss function is convex but not necessarily smooth. First, we propose an accelerated smoothing hard thresholding (ASHT) algorithm for solving such problems, which combines smoothing approximation, extrapolation technique and iterative hard thresholding method. The extrapolation coefficients can be chosen to satisfy \(\sup_k \beta_k=1\). We discuss the convergence of ASHT algorithm with different extrapolation coefficients, and give a sufficient condition to ensure that any accumulation point of the iterates is a local minimizer of the original problem. For a class of special updating schemes on the extrapolation coefficients, we obtain that the iterates are convergent to a local minimizer of the problem, and the convergence rate is \(o(\ln^{\sigma} k/k)\) with \(\sigma \in (1/2, 1]\) on the loss and objective function values. Second, we consider the case in which the loss function is Lipschitz continuously differentiable, and develop an accelerated hard thresholding (AHT) algorithm to solve it. We prove that the iterates of AHT algorithm converge to a local minimizer of the problem that satisfies a desirable lower bound property. Moreover, we show that the convergence rates of loss and objective function values are \(o(k^{-2})\). Finally, some numerical examples are presented to show the theoretical results.A pair of Mond-Weir type third order symmetric dualityhttps://zbmath.org/1522.901942023-12-07T16:00:11.105023Z"Biswal, G."https://zbmath.org/authors/?q=ai:biswal.g"Behera, N."https://zbmath.org/authors/?q=ai:behera.narmada|behera.nikhil-c|behera.nalinikanta|behera.namita|behera.narayan"Mohapatra, R. N."https://zbmath.org/authors/?q=ai:mohapatra.ram-narayan"Padhan, S. K."https://zbmath.org/authors/?q=ai:padhan.saroj-kumarSummary: In this framework, a pair of Mond-Weir type third order symmetric nonlinear programming problems are introduced. Appropriate duality theorems are established for the newly formulated third order symmetric dual problems under the assumptions of boncavity and bonvexity. Different counterexamples are also provided in order to justify the present findings. It is also verified that some of the previously published results in the literarue are particular cases of the findings of the paper.Inexact restoration for minimization with inexact evaluation both of the objective function and the constraintshttps://zbmath.org/1522.901952023-12-07T16:00:11.105023Z"Bueno, L. F."https://zbmath.org/authors/?q=ai:bueno.luis-felipe"Larreal, F."https://zbmath.org/authors/?q=ai:larreal.f"Martínez, J. M."https://zbmath.org/authors/?q=ai:martinez.jose-marioSummary: In a recent paper an Inexact Restoration method for solving continuous constrained optimization problems was analyzed from the point of view of worst-case functional complexity and convergence. On the other hand, the Inexact Restoration methodology was employed, in a different research, to handle minimization problems with inexact evaluation and simple constraints. These two methodologies are combined in the present report, for constrained minimization problems in which both the objective function and the constraints, as well as their derivatives, are subject to evaluation errors. Together with a complete description of the method, complexity and convergence results will be proved.Optimality conditions in DC-constrained mathematical programming problemshttps://zbmath.org/1522.901962023-12-07T16:00:11.105023Z"Correa, Rafael"https://zbmath.org/authors/?q=ai:correa.rafael-augusto-couceiro"López, Marco A."https://zbmath.org/authors/?q=ai:lopez-cerda.marco-antonio"Pérez-Aros, Pedro"https://zbmath.org/authors/?q=ai:perez-aros.pedroSummary: This paper provides necessary and sufficient optimality conditions for abstract-constrained mathematical programming problems in locally convex spaces under new qualification conditions. Our approach exploits the geometrical properties of certain mappings, in particular their structure as difference of convex functions, and uses techniques of generalized differentiation (subdifferential and coderivative). It turns out that these tools can be used fruitfully out of the scope of Asplund spaces. Applications to infinite, stochastic and semi-definite programming are developed in separate sections.Constrained composite optimization and augmented Lagrangian methodshttps://zbmath.org/1522.901972023-12-07T16:00:11.105023Z"De Marchi, Alberto"https://zbmath.org/authors/?q=ai:de-marchi.alberto"Jia, Xiaoxi"https://zbmath.org/authors/?q=ai:jia.xiaoxi"Kanzow, Christian"https://zbmath.org/authors/?q=ai:kanzow.christian"Mehlitz, Patrick"https://zbmath.org/authors/?q=ai:mehlitz.patrickSummary: We investigate finite-dimensional constrained structured optimization problems, featuring composite objective functions and set-membership constraints. Offering an expressive yet simple language, this problem class provides a modeling framework for a variety of applications. We study stationarity and regularity concepts, and propose a flexible augmented Lagrangian scheme. We provide a theoretical characterization of the algorithm and its asymptotic properties, deriving convergence results for fully nonconvex problems. It is demonstrated how the inner subproblems can be solved by off-the-shelf proximal methods, notwithstanding the possibility to adopt any solvers, insofar as they return approximate stationary points. Finally, we describe our matrix-free implementation of the proposed algorithm and test it numerically. Illustrative examples show the versatility of constrained composite programs as a modeling tool and expose difficulties arising in this vast problem class.A Bregman regularized proximal point method for quasi-equilibrium problemshttps://zbmath.org/1522.901982023-12-07T16:00:11.105023Z"Dias, Edimilson L. jun."https://zbmath.org/authors/?q=ai:dias.edimilson-l-jun"Santos, Pedro Jorge S."https://zbmath.org/authors/?q=ai:santos.pedro-jorge-s"de O. Souza, João Carlos"https://zbmath.org/authors/?q=ai:de-o-souza.joao-carlosSummary: We present a Bregman regularized proximal point method for solving quasi-equilibrium problems. Under mild assumptions, we prove that the method finds a solution of the problem and it generalizes some existing works in the literature of equilibrium and quasi-equilibrium problems. A numerical illustration shows the performance of the method in comparison with the classical version of the proximal point method.Manifold mapping: a two-level optimization techniquehttps://zbmath.org/1522.901992023-12-07T16:00:11.105023Z"Echeverría, D."https://zbmath.org/authors/?q=ai:echeverria.david"Hemker, P. W."https://zbmath.org/authors/?q=ai:hemker.pieter-wSummary: In this paper, we analyze in some detail the manifold-mapping optimization technique introduced recently \textit{D. Echeverría} and \textit{P. W. Hemker} [Comput. Methods Appl. Math. 5, No. 2, 107--106 (2005; Zbl 1071.65086)]. Manifold mapping aims at accelerating optimal design procedures that otherwise require many evaluations of time-expensive cost functions. We give a proof of convergence for the manifold-mapping iteration. By means of two simple optimization problems we illustrate the convergence results derived. Finally, the performances of several variants of the method are compared for some design problems from electromagnetics.Convergence properties of an objective-function-free optimization regularization algorithm, including an \(\mathcal{O}(\epsilon^{-3/2})\) complexity boundhttps://zbmath.org/1522.902002023-12-07T16:00:11.105023Z"Gratton, Serge"https://zbmath.org/authors/?q=ai:gratton.serge"Jerad, Sadok"https://zbmath.org/authors/?q=ai:jerad.sadok"Toint, Philippe L."https://zbmath.org/authors/?q=ai:toint.philippe-lSummary: An adaptive regularization algorithm for unconstrained nonconvex optimization is presented in which the objective function is never evaluated but only derivatives are used. This algorithm belongs to the class of adaptive regularization methods, for which optimal worst-case complexity results are known for the standard framework where the objective function is evaluated. It is shown in this paper that these excellent complexity bounds are also valid for the new algorithm despite the fact that significantly less information is used. In particular, it is shown that if derivatives of degree one to \(p\) are used, the algorithm will find an \(\epsilon_1\)-approximate first-order minimizer in at most \(\mathcal{O}(\epsilon_1^{-(p+1)/p})\) iterations and an \((\epsilon_1,\epsilon_2)\)-approximate second-order minimizer in at most \(\mathcal{O}(\max[\epsilon_1^{-(p+1)/p},\epsilon_2^{-(p+1)/(p-1)}])\) iterations. As a special case, the new algorithm using first and second derivatives, when applied to functions with Lipschitz continuous Hessian, will find an iterate \(x_k\) at which the gradient's norm is less than \(\epsilon_1\) in at most \(\mathcal{O}(\epsilon_1^{-3/2})\) iterations.Multilevel objective-function-free optimization with an application to neural networks traininghttps://zbmath.org/1522.902012023-12-07T16:00:11.105023Z"Gratton, Serge"https://zbmath.org/authors/?q=ai:gratton.serge"Kopaničáková, Alena"https://zbmath.org/authors/?q=ai:kopanicakova.alena"Toint, Philippe L."https://zbmath.org/authors/?q=ai:toint.philippe-lSummary: A class of multilevel algorithms for unconstrained nonlinear optimization is presented which does not require the evaluation of the objective function. The class contains the momentum-less AdaGrad method as a particular (single-level) instance. The choice of avoiding the evaluation of the objective function is intended to make the algorithms of the class less sensitive to noise, while the multilevel feature aims at reducing their computational cost. The evaluation complexity of these algorithms is analyzed and their behavior in the presence of noise is then illustrated in the context of training deep neural networks for supervised learning applications.Existence and error bounds of stochastic differential variational inequalitieshttps://zbmath.org/1522.902022023-12-07T16:00:11.105023Z"Guan, Fei"https://zbmath.org/authors/?q=ai:guan.fei"Nguyen, Van Thien"https://zbmath.org/authors/?q=ai:nguyen-van-thien."Peng, Zijia"https://zbmath.org/authors/?q=ai:peng.zijiaSummary: We consider a class of stochastic differential variational inequalities (for short, SDVIs) consisting of a stochastic ordinary differential equation and a stochastic variational inequality. The existence of solutions to SDVIs is proved under two cases that the leading operator in the stochastic variational inequality is \(P\)-function and \(P_0\)-function. Then, the least-norm solution to the second case is obtained by a regularized method. Moreover, the mean square convergence and error bounds of the time-stepping method to SDVIs are established.A new hybrid conjugate gradient algorithm for unconstrained optimizationhttps://zbmath.org/1522.902032023-12-07T16:00:11.105023Z"Hafaidia, Imane"https://zbmath.org/authors/?q=ai:hafaidia.imane"Guebbai, Hamza"https://zbmath.org/authors/?q=ai:guebbai.hamza"Al-Baali, Mehiddin"https://zbmath.org/authors/?q=ai:al-baali.mehiddin"Ghiat, Mourad"https://zbmath.org/authors/?q=ai:ghiat.mouradSummary: It is well known that conjugate gradient methods are useful for solving large-scale unconstrained nonlinear optimization problems. In this paper, we consider combining the best features of two conjugate gradient methods. In particular, we give a new conjugate gradient method, based on the hybridization of the useful DY (Dai-Yuan), and HZ (Hager-Zhang) methods. The hybrid parameters are chosen such that the proposed method satisfies the conjugacy and sufficient descent conditions. It is shown that the new method maintains the global convergence property of the above two methods. The numerical results are described for a set of standard test problems. It is shown that the performance of the proposed method is better than that of the DY and HZ methods in most cases.Convergence rate estimates for penalty methods revisitedhttps://zbmath.org/1522.902042023-12-07T16:00:11.105023Z"Izmailov, A. F."https://zbmath.org/authors/?q=ai:izmailov.aleksei-f"Solodov, M. V."https://zbmath.org/authors/?q=ai:solodov.mikhail-vSummary: For the classical quadratic penalty, it is known that the distance from the solution of the penalty subproblem to the solution of the original problem is at worst inversely proportional to the value of the penalty parameter under the linear independence constraint qualification, strict complementarity, and the second-order sufficient optimality conditions. Moreover, using solutions of the penalty subproblem, one can obtain certain useful Lagrange multipliers estimates whose distance to the optimal ones is also at least inversely proportional to the value of the parameter. We show that the same properties hold more generally, namely, under the (weaker) strict Mangasarian-Fromovitz constraint qualification and second-order sufficiency (and without strict complementarity). Moreover, under the linear independence constraint qualification and strong second-order sufficiency (also without strict complementarity), we demonstrate local uniqueness and Lipschitz continuity of stationary points of penalty subproblems. In addition, those results follow from the analysis of general power penalty functions, of which quadratic penalty is a special case.Optimization strategies for the bilevel network design problem with affine cost functionshttps://zbmath.org/1522.902052023-12-07T16:00:11.105023Z"Krylatov, Alexander"https://zbmath.org/authors/?q=ai:krylatov.aleksandr-yu"Raevskaya, Anastasiya"https://zbmath.org/authors/?q=ai:raevskaya.anastasiya-p"Ageev, Petr"https://zbmath.org/authors/?q=ai:ageev.petrSummary: Today artificial intelligence systems support efficient management in different fields of social activities. In particular, congestion control in modern networks seems to be impossible without proper mathematical models of traffic flow assignment. Thus, the network design problem can be referred to as Stackelberg game with independent lower-level drivers acting in a non-cooperative manner to minimize individual costs. In turn, upper-level decision-maker seeks to minimize overall travel time in the network by investing in its capacity. Hence, the decision-maker faces the challenge with a hierarchical structure which solution important due to its influence on the sustainable development of modern large cities. However, it is well-known that a bilevel programming problem is often strongly NP-hard, while the hierarchical optimization structure of a bilevel problem raises such difficulties as non-convexity and discontinuity. The present paper is devoted to the network design problem with affine cost functions. First of all, we obtain exact optimality conditions to the network design problem in a case of a single-commodity network with non-interfering routes. Secondly, we show that obtained conditions can be exploited as optimization strategies for solving the network design problem in the case of an arbitrary network with affine cost functions. These findings give fresh managerial insights to traffic engineers dealing with network design.Incorporating multiple a priori information for inverse problem by inexact scaled gradient projectionhttps://zbmath.org/1522.902062023-12-07T16:00:11.105023Z"Li, Da"https://zbmath.org/authors/?q=ai:li.da"Lamoureux, Michael P."https://zbmath.org/authors/?q=ai:lamoureux.michael-p"Liao, Wenyuan"https://zbmath.org/authors/?q=ai:liao.wenyuanSummary: Many inverse problems can be formulated as constrained optimization problems in which the feasible set is a description of the a priori information. Different feasible sets represent different a priori information. In this work, we investigate the constrained optimization problem such that the feasible set is the intersection of several convex sets. A closed-form projection is necessary when using standard gradient projection methods to solve the constrained optimization problem since the projection has to be evaluated exactly. However, the numerical methods of projection onto the intersection of convex sets usually have an iterative structure which leads to an inexact projection result in practice. In this work, a feasible set expanding strategy is designed to overcome this issue. Based on this strategy, an inexact scaled gradient projection method is proposed. The convergence analysis is provided with proper assumptions. Numerical results for an inverse problem named full waveform inversion are provided.A new nonmonotone spectral projected gradient algorithm for box-constrained optimization problems in \(m \times n\) real matrix space with application in image clusteringhttps://zbmath.org/1522.902072023-12-07T16:00:11.105023Z"Li, Ting"https://zbmath.org/authors/?q=ai:li.ting"Wan, Zhong"https://zbmath.org/authors/?q=ai:wan.zhong"Guo, Jie"https://zbmath.org/authors/?q=ai:guo.jieSummary: Box-constrained optimization problems in the real \(m \times n\) matrix space have been widely applied in big data mining. However, efficient solution of them is still a challenge. In this paper, a new nonmonotone line search rule is first proposed by extending the well-known ones and inheriting their advantages. Then, by analyzing and exploiting properties of this rule, a new nonmonotone spectral projected gradient algorithm is developed to solve the box-constrained optimization problems in the matrix space. Global convergence of the developed algorithm is also established. Numerical tests are conducted on a series of randomly generated test problems and those in the set of benchmark test problems. Compared with other existing nonmonotone line search rules, our rule shows its advantages in terms of the significantly reduced number of function evaluations and significantly reduced number of iterations. To further validate applicability of this research, we apply the studied optimization problem and the developed algorithm to solve the problems of image clustering. Numerical results demonstrate that the proposed method can generate better clustering results and is more robust than the similar ones available in the literature.Variational analysis of norm cones in finite dimensional Euclidean spaceshttps://zbmath.org/1522.902082023-12-07T16:00:11.105023Z"Liu, Haoyang"https://zbmath.org/authors/?q=ai:liu.haoyang"Wu, Jia"https://zbmath.org/authors/?q=ai:wu.jia"Zhang, Liwei"https://zbmath.org/authors/?q=ai:zhang.liweiSummary: A norm cone in a finite dimensional Euclidean space is the epigraph of a norm. Many important practical optimization problems are formulated as norm conic optimization problems, a typical example is the second-order conic optimization problem. This paper is devoted to the study of variational analysis of norm cones. For a general norm cone, formulas for the tangent cone, normal cone and second-order tangent set are derived. The projection on the norm cone and its differentiability properties are developed, including formulas for derivatives, directional derivatives and B-subdifferentials. All these results are specified to the \(l_p\)-norm, \(p \in (1,\infty)\). Additionally, variational geometries of \(l_1\)-norm cone and \(l_{\infty}\)-norm cone are characterized.Smoothing neural network for \(L_0\) regularized optimization problem with general convex constraintshttps://zbmath.org/1522.902092023-12-07T16:00:11.105023Z"Li, Wenjing"https://zbmath.org/authors/?q=ai:li.wenjing.1|li.wenjing"Bian, Wei"https://zbmath.org/authors/?q=ai:bian.weiSummary: In this paper, we propose a neural network modeled by a differential inclusion to solve a class of discontinuous and nonconvex sparse regression problems with general convex constraints, whose objective function is the sum of a convex but not necessarily differentiable loss function and \(L_0\) regularization. We construct a smoothing relaxation function of \(L_0\) regularization and propose a neural network to solve the considered problem. We prove that the solution of proposed neural network with any initial point satisfying linear equality constraints is global existent, bounded and reaches the feasible region in finite time and remains there thereafter. Moreover, the solution of proposed neural network is its slow solution and any accumulation point of it is a Clarke stationary point of the brought forward nonconvex smoothing approximation problem. In the box-constrained case, all accumulation points of the solution own a unified lower bound property and have a common support set. Except for a special case, any accumulation point of the solution is a local minimizer of the considered problem. In particular, the proposed neural network has a simple structure than most existing neural networks for solving the locally Lipschitz continuous but nonsmooth nonconvex problems. Finally, we give some numerical experiments to show the efficiency of proposed neural network.On local error bound in nonlinear programshttps://zbmath.org/1522.902102023-12-07T16:00:11.105023Z"Minchenko, L. I."https://zbmath.org/authors/?q=ai:minchenko.leonid-ivanovich"Sirotko, S. I."https://zbmath.org/authors/?q=ai:sirotko.sergei-ivanovichSummary: Numerous efforts in the literature are devoted to studying error bounds in optimization problems. The existence of local error bounds is closely related with constraint qualifications. It is well-known that some constraint qualifications imply error bounds. We consider constraint qualifications, which do not impose as strong requirements on the structure of the optimization problem as traditional conditions do. On their base necessary and sufficient conditions of error bounds are derived.
For the entire collection see [Zbl 1508.90001].Nonlinear fuzzy fractional signomial programming problem: a fuzzy geometric programming solution approachhttps://zbmath.org/1522.902112023-12-07T16:00:11.105023Z"Mishra, Sudipta"https://zbmath.org/authors/?q=ai:mishra.sudipta"Ota, Rashmi Ranjan"https://zbmath.org/authors/?q=ai:ota.rashmi-ranjan"Nayak, Suvasis"https://zbmath.org/authors/?q=ai:nayak.suvasisSummary: Fuzzy fractional signomial programming problem is a relatively new optimization problem. In real world problems, some variables may vacillate because of various reasons. To tackle these vacillating variables, vagueness is considered in form of fuzzy sets. In this paper, a nonlinear fuzzy fractional signomial programming problem is considered with all its coefficients in objective functions as well as constraints are fuzzy numbers. Two solution approaches are developed based on signomial geometric programming comprising nearest interval approximation with parametric interval valued functions and fuzzy \(\alpha \)-cut with min-max approach. To demonstrate the proposed methods, two illustrative numerical examples are solved and the results are comparatively discussed showing its feasibility and effectiveness.Dinkelbach type approximation algorithms for nonlinear fractional optimization problemshttps://zbmath.org/1522.902122023-12-07T16:00:11.105023Z"Orzan, Alexandru"https://zbmath.org/authors/?q=ai:orzan.alexandru"Precup, Radu"https://zbmath.org/authors/?q=ai:precup.radu|precup.radu-emilSummary: In this paper we establish some approximation versions of the classical Dinkelbach algorithm for nonlinear fractional optimization problems in Banach spaces. We start by examining what occurs if at any step of the algorithm, the generated point desired to be a minimizer can only be determined with a given error. Next we assume that the step error tends to zero as the algorithm advances. The last version of the algorithm we present is making use of Ekeland's variational principle for generating the sequence of minimizer-like points. In the final part of the article we deliver some results in order to achieve a Palais-Smale type compactness condition that guarantees the convergence of our Dinkelbach-Ekeland algorithm.Constraint qualifications in terms of convexificators for nonsmooth programming problems with mixed constraintshttps://zbmath.org/1522.902132023-12-07T16:00:11.105023Z"Rimpi"https://zbmath.org/authors/?q=ai:rimpi."Lalitha, C. S."https://zbmath.org/authors/?q=ai:lalitha.c-sSummary: The main aim of the paper is to introduce certain constraint qualifications for a nonsmooth programming problem in terms of semi-regular convexificators and investigate their relations with other existing notions of constraint qualifications. The programming problem under consideration has mixed constraints, that is, it involves both inequality and equality constraints. All these notions are in terms of upper semi-regular convexificators of inequality constraints and pseudo-differentials of equality constraints. Based on a sufficient condition for error bound property, the implication relation between quasinormality and error bound property in terms of convexificators is investigated in this paper. Three conditions are introduced, namely constant positive linear dependence condition (CPLD), constant rank constraint qualification (CRCQ) and Mangasarian-Fromovitz constraint qualification (MFCQ) in terms of convexificators. These conditions are in fact shown to be constraint qualifications as Karush-Kuhn-Tucker optimality conditions hold when CPLD holds and both MFCQ and CRCQ imply CPLD. Further, it is observed that CPLD and quasinormality conditions are independent for nonsmooth problems in terms of convexificators.A nonlinear conjugate gradient method using inexact first-order informationhttps://zbmath.org/1522.902142023-12-07T16:00:11.105023Z"Zhao, Tiantian"https://zbmath.org/authors/?q=ai:zhao.tiantian"Yang, Wei Hong"https://zbmath.org/authors/?q=ai:yang.weihongSummary: Conjugate gradient methods are widely used for solving nonlinear optimization problems. In some practical problems, we can only get approximate values of the objective function and its gradient. It is necessary to consider optimization algorithms that use inexact function evaluations and inexact gradients. In this paper, we propose an inexact nonlinear conjugate gradient (INCG) method to solve such problems. Under some mild conditions, the global convergence of INCG is proved. Specifically, we establish the linear convergence of INCG when the objective function is strongly convex. Numerical results demonstrate that, compared to the state-of-the-art algorithms, INCG is an effective method.A semismooth Newton based augmented Lagrangian method for nonsmooth optimization on matrix manifoldshttps://zbmath.org/1522.902152023-12-07T16:00:11.105023Z"Zhou, Yuhao"https://zbmath.org/authors/?q=ai:zhou.yuhao"Bao, Chenglong"https://zbmath.org/authors/?q=ai:bao.chenglong"Ding, Chao"https://zbmath.org/authors/?q=ai:ding.chao"Zhu, Jun"https://zbmath.org/authors/?q=ai:zhu.jun.2|zhu.junSummary: This paper is devoted to studying an augmented Lagrangian method for solving a class of manifold optimization problems, which have nonsmooth objective functions and nonlinear constraints. Under the constant positive linear dependence condition on manifolds, we show that the proposed method converges to a stationary point of the nonsmooth manifold optimization problem. Moreover, we propose a globalized semismooth Newton method to solve the augmented Lagrangian subproblem on manifolds efficiently. The local superlinear convergence of the manifold semismooth Newton method is also established under some suitable conditions. We also prove that the semismoothness on submanifolds can be inherited from that in the ambient manifold. Finally, numerical experiments on compressed modes and (constrained) sparse principal component analysis illustrate the advantages of the proposed method.Lipschitz upper semicontinuity in linear optimization via local directional convexityhttps://zbmath.org/1522.902162023-12-07T16:00:11.105023Z"Camacho, J."https://zbmath.org/authors/?q=ai:camacho.james-jun|camacho.jorge-fernando|camacho.jose-r|camacho.juan|camacho.jan-tracy|camacho.jose-carlos"Cánovas, M. J."https://zbmath.org/authors/?q=ai:canovas.maria-josefa"Parra, J."https://zbmath.org/authors/?q=ai:parra.juanSummary: This work is focussed on computing the Lipschitz upper semicontinuity modulus of the argmin mapping for canonically perturbed linear programs. The immediate antecedent can be traced out from [\textit{J. Camacho} et al., SIAM J. Optim. 32, No. 4, 2859--2878 (2022; Zbl 1507.90176)], devoted to the feasible set mapping. The aimed modulus is expressed in terms of a finite amount of calmness moduli, previously studied in the literature. Despite the parallelism in the results, the methodology followed in the current paper differs notably from [Camacho et al., loc. cit.] as far as the graph of the argmin mapping is not convex; specifically, a new technique based on a certain type of local directional convexity is developed.Optimal parameters for numerical solvers of PDEshttps://zbmath.org/1522.902172023-12-07T16:00:11.105023Z"Frasca-Caccia, Gianluca"https://zbmath.org/authors/?q=ai:frasca-caccia.gianluca"Singh, Pranav"https://zbmath.org/authors/?q=ai:singh.pranavSummary: In this paper we introduce a procedure for identifying optimal methods in parametric families of numerical schemes for initial value problems in partial differential equations. The procedure maximizes accuracy by adaptively computing optimal parameters that minimize a defect-based estimate of the local error at each time step. Viable refinements are proposed to reduce the computational overheads involved in the solution of the optimization problem, and to maintain conservation properties of the original methods. We apply the new strategy to recently introduced families of conservative schemes for the Korteweg-de Vries equation and for a nonlinear heat equation. Numerical tests demonstrate the improved efficiency of the new technique in comparison with existing methods.On stability of maximal entropy OWA operator weightshttps://zbmath.org/1522.902182023-12-07T16:00:11.105023Z"Harmati, István Á."https://zbmath.org/authors/?q=ai:harmati.istvan-a"Fullér, Robert"https://zbmath.org/authors/?q=ai:fuller.robert-c-g|fuller.robert-w"Felde, Imre"https://zbmath.org/authors/?q=ai:felde.imreSummary: The maximal entropy OWA operator (MEOWA) weights can be obtained by solving a nonlinear programming problem with a linear constraint for the level of orness. Since the exact MEOWA weights are not known for the general case we can only find approximate solutions. We will prove that the nonlinear programming problem for obtaining MEOWA weights is well-posed: it has a unique solution and each MEOWA weight changes continuously with the initial level of orness. Using the implicit function theorem we will show that MEOWA weights are Lipschitz-continuous functions of the orness level. The stability property of the MEOWA weights under small changes of the orness level guarantees that small rounding errors of digital computation and small errors of measurement of the orness level can cause only a small deviation in MEOWA weights, i.e. every successive approximation method can be applied to the computation of the approximation of the exact MEOWA weights.Sensitivity analysis of combinatorial optimization problems using evolutionary bilevel optimization and data mininghttps://zbmath.org/1522.902192023-12-07T16:00:11.105023Z"Schulte, Julian"https://zbmath.org/authors/?q=ai:schulte.julian"Nissen, Volker"https://zbmath.org/authors/?q=ai:nissen.volkerSummary: Sensitivity analysis in general deals with the question of how changes in input data of a model affect its output data. In the context of optimization problems, such an analysis could, for instance, address how changes in capacity constraints affect the optimal solution value. Although well established in the domain of linear programming, sensitivity analysis approaches for combinatorial optimization problems are model-specific, limited in scope and not applicable to practical optimization problems. To overcome these limitations, Schulte et al. developed the concept of bilevel innovization. By using evolutionary bilevel optimization in combination with data mining and visualization techniques, bilevel innovization provides decision-makers with deeper insights into the behavior of the optimization model and supports decision-making related to model building and configuration. Originally introduced in the field of evolutionary computation, most recently bilevel innovization has been proposed as an approach to sensitivity analysis for combinatorial problems in general. Based on previous work on bilevel innovization, our paper illustrates this concept as a tool for sensitivity analysis by providing a comprehensive analysis of the generalized assignment problem. Furthermore, it is investigated how different algorithms for solving the combinatorial problem affect the insights gained by the sensitivity analysis, thus evaluating the robustness and reliability of the sensitivity analysis results.Primal characterizations of error bounds for composite-convex inequalitieshttps://zbmath.org/1522.902202023-12-07T16:00:11.105023Z"Wei, Zhou"https://zbmath.org/authors/?q=ai:wei.zhou"Théra, Michel"https://zbmath.org/authors/?q=ai:thera.michel-a"Yao, Jen-Chih"https://zbmath.org/authors/?q=ai:yao.jen-chihSummary: This paper is devoted to primal conditions of error bounds for a general function. In terms of Bouligand tangent cones, lower Hadamard directional derivatives and the Hausdorff-Pompeiu excess of subsets, we provide several necessary and/or sufficient conditions for error bounds with mild assumptions. Then we use these primal results to characterize error bounds for composite-convex functions (i.e. the composition of a convex function with a continuously differentiable mapping). It is proved that the primal characterization of error bounds can be established via Bouligand tangent cones, directional derivatives and the Hausdorff-Pompeiu excess if the mapping is metrically regular at the given point. The accurate estimate on the error bound modulus is also obtained.Robust approach for uncertain multi-dimensional fractional control optimization problemshttps://zbmath.org/1522.902212023-12-07T16:00:11.105023Z"Jayswal, Anurag"https://zbmath.org/authors/?q=ai:jayswal.anurag"Baranwal, Ayushi"https://zbmath.org/authors/?q=ai:baranwal.ayushiSummary: In this paper, we focus our study on a multi-dimensional fractional control optimization problem involving data uncertainty (FP) and derive the parametric robust necessary optimality conditions and its sufficiency by imposing the convexity hypotheses on the involved functionals. We also construct the parametric robust dual problem associated with the above-considered problem (FP) and establish the weak and strong robust duality theorems. The strong robust duality theorem asserts that the duality gap is zero under the convexity notion. In addition, we formulate some examples to validate the stated conclusions.Efficient algorithm for globally computing the min-max linear fractional programming problemhttps://zbmath.org/1522.902222023-12-07T16:00:11.105023Z"Jiao, Hongwei"https://zbmath.org/authors/?q=ai:jiao.hongwei"Wang, Wenjie"https://zbmath.org/authors/?q=ai:wang.wenjie"Ge, Li"https://zbmath.org/authors/?q=ai:ge.li"Shen, Peiping"https://zbmath.org/authors/?q=ai:shen.peiping"Shang, Youlin"https://zbmath.org/authors/?q=ai:shang.youlinSummary: In this paper, we consider the min-max linear fractional programming problem (MLFP) which is NP-hard. We first introduce some auxiliary variables to derive an equivalent problem of the problem (MLFP). An outer space branch-and-bound algorithm is then designed by integrating some basic operations such as the linear relaxation method and branching rule. The global convergence of the proposed algorithm is proved by means of the subsequent solutions of a series of linear relaxation programming problems, and the computational complexity of the proposed algorithm is estimated based on the branching rule. Finally, numerical experimental results demonstrate the proposed algorithm can be used to efficiently compute the globally optimal solutions of test examples.On optimality conditions and duality for multiobjective optimization with equilibrium constraintshttps://zbmath.org/1522.902232023-12-07T16:00:11.105023Z"Khanh, P. Q."https://zbmath.org/authors/?q=ai:phan-quoc-khanh."Tung, L. T."https://zbmath.org/authors/?q=ai:tung.le-thanhSummary: In this paper, we consider nonsmooth multiobjective optimization problems with equilibrium constraints. Necessary/sufficient conditions for optimality in terms of the Michel-Penot subdifferential are established. Then, we propose Wolfe- and Mond-Weir-types of dual problems and investigate duality relations under generalized convexity assumptions. Some examples are provided to illustrate our results.On optimality and duality for second-order cone linear fractional optimization problemshttps://zbmath.org/1522.902242023-12-07T16:00:11.105023Z"Kim, Gwi Soo"https://zbmath.org/authors/?q=ai:kim.gwi-soo"Kim, Moon Hee"https://zbmath.org/authors/?q=ai:kim.moon-hee"Lee, Gue Myung"https://zbmath.org/authors/?q=ai:lee.gue-myungSummary: We consider a second-order linear fractional optimization problem (P), and obtain optimality conditions for (P) which hold without any constraint qualification. Moreover, we formulate the nonfractional dual problem of (P) and then prove the duality theorem (weak duality theorem, strong duality theorem). The strong duality theorem holds without any constraint qualification.
For the entire collection see [Zbl 1480.00062].A trust-region LP-Newton method for constrained nonsmooth equations under Hölder metric subregularityhttps://zbmath.org/1522.902252023-12-07T16:00:11.105023Z"Becher, Letícia"https://zbmath.org/authors/?q=ai:becher.leticia"Fernández, Damián"https://zbmath.org/authors/?q=ai:fernandez.damian"Ramos, Alberto"https://zbmath.org/authors/?q=ai:ramos.alberto-gil-c-p|ramos.alberto.2|ramos.albertoSummary: We describe and analyze a globally convergent algorithm to find a possible nonisolated zero of a piecewise smooth mapping over a polyhedral set. Such formulation includes Karush-Kuhn-Tucker systems, variational inequalities problems, and generalized Nash equilibrium problems. Our algorithm is based on a modification of the fast locally convergent Linear Programming (LP)-Newton method with a trust-region strategy for globalization that makes use of the natural merit function. The transition between global and local convergence occurs naturally under mild assumption. Our local convergence analysis of the method is performed under a Hölder metric subregularity condition of the mapping defining the possibly nonsmooth equation and the Hölder continuity of the derivative of the selection mapping. We present numerical results that show the feasibility of the approach.On a threshold descent method for quasi-equilibriahttps://zbmath.org/1522.902262023-12-07T16:00:11.105023Z"Bianchi, M."https://zbmath.org/authors/?q=ai:bianchi.monica"Konnov, I."https://zbmath.org/authors/?q=ai:konnov.igor-v"Pini, R."https://zbmath.org/authors/?q=ai:pini.ritaSummary: We describe a special class of quasi-equilibrium problems in metric spaces and propose a novel simple threshold descent method for solving these problems. Due to the framework, the convergence of the method cannot be established with the usual convexity or generalized convexity assumptions. Under mild conditions, the iterative procedure gives solutions of the quasi-equilibrium problem. We apply this method to scalar and vector generalized quasi-equilibrium problems and to some classes of relative optimization problems.A path-following interior-point algorithm for monotone LCP based on a modified Newton search directionhttps://zbmath.org/1522.902272023-12-07T16:00:11.105023Z"Grimes, Welid"https://zbmath.org/authors/?q=ai:grimes.welid"Achache, Mohamed"https://zbmath.org/authors/?q=ai:achache.mohamedSummary: In this paper, we propose a short-step feasible full-Newton step path-following interior-point algorithm (IPA) for monotone linear complementarity problems (LCPs). The proposed IPA uses the technique of algebraic equivalent transformation (AET) induced by an univariate function to transform the centering equations which defines the central-path. By applying Newton's method to the modified system of the central-path of LCP, a new Newton search direction is obtained. Under new appropriate defaults of the threshold \(\tau\) which defines the size of the neighborhood of the central-path and of \(\theta\) which determines the decrease in the barrier parameter, we prove that the IPA is well-defined and converges locally quadratically to a solution of the monotone LCPs. Moreover, we derive its iteration bound, namely, \(\mathcal{O}(\sqrt{n}\log\frac{n}{\varepsilon})\) which coincides with the best-known iteration bound for such algorithms. Finally, some numerical results are presented to show its efficiency.Distributionally robust expected residual minimization for stochastic variational inequality problemshttps://zbmath.org/1522.902282023-12-07T16:00:11.105023Z"Hori, Atsushi"https://zbmath.org/authors/?q=ai:hori.atsushi"Yamakawa, Yuya"https://zbmath.org/authors/?q=ai:yamakawa.yuya"Yamashita, Nobuo"https://zbmath.org/authors/?q=ai:yamashita.nobuoSummary: The stochastic variational inequality problem (SVIP) is an equilibrium model that includes random variables and has been widely applied in various fields such as economics and engineering. Expected residual minimization (ERM) is an established model for obtaining a reasonable solution for the SVIP, and its objective function is an expected value of a suitable merit function for the SVIP. However, the ERM is restricted to the case where the distribution is known in advance. We extend the ERM to ensure the attainment of robust solutions for the SVIP under the uncertainty distribution (the extended ERM is referred to as distributionally robust expected residual minimization (DRERM), where the worst-case distribution is derived from the set of probability measures in which the expected value and variance take the same sample mean and variance, respectively). Under suitable assumptions, we demonstrate that the DRERM can be reformulated as a deterministic convex nonlinear semidefinite programming to avoid numerical integration.A smoothing Newton method based on the modulus equation for a class of weakly nonlinear complementarity problemshttps://zbmath.org/1522.902292023-12-07T16:00:11.105023Z"Huang, Baohua"https://zbmath.org/authors/?q=ai:huang.baohua"Li, Wen"https://zbmath.org/authors/?q=ai:li.wen.1Summary: By equivalently transforming a class of weakly nonlinear complementarity problems into a modulus equation, and introducing a smoothing approximation of the absolute value function, a smoothing Newton method is established for solving the weakly nonlinear complementarity problem. Under some mild assumptions, the proposed method is shown to possess global convergence and locally quadratical convergence. Especially, the global convergence results do not need a priori existence of an accumulation point with some suitable conditions. Numerical results are given to show the efficiency of the proposed method.A fixed point iterative method for tensor complementarity problems with the implicit \(Z\)-tensorshttps://zbmath.org/1522.902302023-12-07T16:00:11.105023Z"Huang, Zheng-Hai"https://zbmath.org/authors/?q=ai:huang.zheng-hai"Li, Yu-Fan"https://zbmath.org/authors/?q=ai:li.yufan"Wang, Yong"https://zbmath.org/authors/?q=ai:wang.yong.13Summary: In this paper, we consider solving the tensor complementarity problem (TCP). We first introduce the concept of the implicit \(Z\)-tensor, which is a generalization of \(Z\)-tensor. Then, based on a new fixed point reformulation of the TCP, we design an iterative algorithm for solving the TCP with an implicit \(Z\)-tensor under the assumption that the feasible set of the problem involved is nonempty. We prove that the proposed fixed point iterative method converges monotonically downward to a solution of the TCP. Furthermore, we establish the global linear rate of convergence of the proposed method under some reasonable assumptions. Compared with the existing related studies, the proposed method not only solves a wider range of TCPs, but also has a lower computational cost. The numerical results verify our theoretical findings.The CQ projection method and equilibrium problems in Hadamard spaceshttps://zbmath.org/1522.902312023-12-07T16:00:11.105023Z"Kimura, Yasunori"https://zbmath.org/authors/?q=ai:kimura.yasunori"Naiki, Sota"https://zbmath.org/authors/?q=ai:naiki.sotaSummary: We consider equilibrium problems in Hadamard spaces and prove the convergence of an iterative sequence generated by the CQ projection method.
For the entire collection see [Zbl 1480.00062].Sufficient conditions for perturbations to define the resolvent of the equilibrium problem on complete geodesic spaceshttps://zbmath.org/1522.902322023-12-07T16:00:11.105023Z"Kimura, Yasunori"https://zbmath.org/authors/?q=ai:kimura.yasunori"Sasaki, Kazuya"https://zbmath.org/authors/?q=ai:sasaki.kazuyaSummary: In this paper, we consider sufficient conditions for perturbations to define a resolvent of the equilibrium problem on complete geodesic spaces with curvature bounded above.Regularized equilibrium problems with equilibrium constraints with application to energy marketshttps://zbmath.org/1522.902332023-12-07T16:00:11.105023Z"Luna, Juan Pablo"https://zbmath.org/authors/?q=ai:luna.juan-pablo"Sagastizábal, Claudia"https://zbmath.org/authors/?q=ai:sagastizabal.claudia-a"Filiberti, Julia"https://zbmath.org/authors/?q=ai:filiberti.julia"Gabriel, Steven A."https://zbmath.org/authors/?q=ai:gabriel.steven-a"Solodov, Mikhail V."https://zbmath.org/authors/?q=ai:solodov.mikhail-vSummary: Equilibrium problems with equilibrium constraints are appropriate modeling formulations in a number of important areas, such as energy markets, transportation planning, and logistics. These models often correspond to bilevel games, in which certain dual variables, representing the equilibrium price, play a fundamental role. We consider multileader single-follower equilibrium problems having a linear program in the lower level. Because in this setting the lower-level response to the leaders' decisions may not be unique, the game formulation becomes ill-posed. We resolve possible ambiguities by considering a sequence of bilevel equilibrium problems, endowed with a special regularization term. We prove convergence of the approximating scheme. Our technique proves useful numerically over several instances related to energy markets. When using PATH to solve the corresponding mixed-complementarity formulations, we exhibit that, in the given context, the regularization approach computes a genuine equilibrium price almost always, while without regularization the outcome is quite the opposite.Structured tensor tuples to polynomial complementarity problemshttps://zbmath.org/1522.902342023-12-07T16:00:11.105023Z"Shang, Tong-tong"https://zbmath.org/authors/?q=ai:shang.tong-tong"Tang, Guo-ji"https://zbmath.org/authors/?q=ai:tang.guojiSummary: It is well known that structured tensors play an important role in the investigation of tensor complementarity problems. The polynomial complementarity problem is a natural generalization of the tensor complementarity problem. Similar to the investigation of tensor complementarity problems, it is believed that structured tensor tuples will play an important role in the investigation of polynomial complementarity problems. In the present paper, several classes of structured tensor tuples are introduced and the relationships between them are discussed. By using the structured tensor(s) (tuples), the uniqueness of the solution and the global upper bound of the solution set of the polynomial complementarity problem are investigated. The results presented in the present paper generalize the corresponding those in the recent literature.On local uniqueness of normalized Nash equilibriahttps://zbmath.org/1522.902352023-12-07T16:00:11.105023Z"Shikhman, Vladimir"https://zbmath.org/authors/?q=ai:shikhman.vladimirSummary: For generalized Nash equilibrium problems (GNEP) with shared constraints we focus on the notion of normalized Nash equilibrium in the nonconvex setting. The property of nondegeneracy for normalized Nash equilibria is introduced. Nondegeneracy refers to GNEP-tailored versions of linear independence constraint qualification, strict complementarity and second-order regularity. Surprisingly enough, nondegeneracy of normalized Nash equilibrium does not prevent from degeneracies at the individual players' level. We show that generically all normalized Nash equilibria are nondegenerate. Moreover, nondegeneracy turns out to be a sufficient condition for the local uniqueness of normalized Nash equilibria. We emphasize that even in the convex setting the proposed notion of nondegeneracy differs from the sufficient condition for (global) uniqueness of normalized Nash equilbira, which is known from the literature.A sequential ADMM algorithm to find sparse LCP solutions using a \(l_2-l_1\) regularization technique with application in bimatrix gamehttps://zbmath.org/1522.902362023-12-07T16:00:11.105023Z"Wang, Feiran"https://zbmath.org/authors/?q=ai:wang.feiran"Yuan, Yuan"https://zbmath.org/authors/?q=ai:yuan.yuan.2|yuan.yuan.1|yuan.yuan|yuan.yuan.3"Wang, Xiaoquan"https://zbmath.org/authors/?q=ai:wang.xiaoquan"Shao, Hu"https://zbmath.org/authors/?q=ai:shao.hu"Shen, Liang"https://zbmath.org/authors/?q=ai:shen.liangSummary: The linear complementary problem (LCP) is a unified formulation for linear and quadratic programming problems. Therefore, it has many applications in practical problems like bimatrix game. We prove that it makes sense to look for sparse LCP solutions. A \(l_2-l_1\) regularization technique transforms the original sparse optimization problem into an unconstrained one. Thereafter, a linearized ADMM (for alternating direction method of multipliers) is designed to solve the regularization model. Then, using a penalty function approach, we propose an efficient sequential linearized ADMM to find the sparse LCP solutions. Finally, numerical experiments prove that the sparse solution of LCPs can be solved efficiently, and is competitive with other state-of-the-art algorithms. A practical application in bimatrix game is also reported.A modulus-based matrix splitting method for the vertical nonlinear complementarity problemhttps://zbmath.org/1522.902372023-12-07T16:00:11.105023Z"Xie, Shuilian"https://zbmath.org/authors/?q=ai:xie.shuilian"Yang, Zhen-Ping"https://zbmath.org/authors/?q=ai:yang.zhenping"Xu, Hongru"https://zbmath.org/authors/?q=ai:xu.hongruSummary: Many applications arising in control theory, nonlinear networks, generalized Leontief input-output model and economics lead to a broad range of optimization and equilibrium problems. Under suitable convexity assumptions, the equilibrium conditions of such problems may be compactly stated as the vertical nonlinear complementarity problem (VNCP). In this paper, based on modulus-based formulation of the VNCP, we present a variety of modulus-based matrix splitting methods for solving the VNCP. Under some mild conditions, we establish the convergence of the proposed method. Numerical experiments indicate that the proposed method is effective.Recent advances in multiobjective convex semi-infinite optimizationhttps://zbmath.org/1522.902382023-12-07T16:00:11.105023Z"Goberna, M."https://zbmath.org/authors/?q=ai:goberna.miguel-angel"Todorov, M. I."https://zbmath.org/authors/?q=ai:ivanov-todorov.maximSummary: This paper reviews the existing literature on multiobjective (or vector) semi-infinite optimization problems, which are defined by finitely many convex objective functions of finitely many variables whose feasible sets are described by infinitely many convex constraints. The paper shows several applications of this type of optimization problems and presents a state-of-the-art review of its methods and theoretical developments (in particular, optimality, duality, and stability).Error bounds for discrete-continuous free flight trajectory optimizationhttps://zbmath.org/1522.902392023-12-07T16:00:11.105023Z"Borndörfer, Ralf"https://zbmath.org/authors/?q=ai:borndorfer.ralf"Danecker, Fabian"https://zbmath.org/authors/?q=ai:danecker.fabian"Weiser, Martin"https://zbmath.org/authors/?q=ai:weiser.martinSummary: Two-stage methods addressing continuous shortest path problems start local minimization from discrete shortest paths in a spatial graph. The convergence of such hybrid methods to global minimizers hinges on the discretization error induced by restricting the discrete global optimization to the graph, with corresponding implications on choosing an appropriate graph density. A prime example is flight planning, i.e., the computation of optimal routes in view of flight time and fuel consumption under given weather conditions. Highly efficient discrete shortest path algorithms exist and can be used directly for computing starting points for locally convergent optimal control methods. We derive a priori and localized error bounds for the flight time of discrete paths relative to the optimal continuous trajectory, in terms of the graph density and the given wind field. These bounds allow designing graphs with an optimal local connectivity structure. The properties of the bounds are illustrated on a set of benchmark problems. It turns out that localization improves the error bound by four orders of magnitude, but still leaves ample opportunities for tighter error bounds by a posteriori estimators.Additional closeness of cycle graphshttps://zbmath.org/1522.902402023-12-07T16:00:11.105023Z"Dangalchev, Chavdar"https://zbmath.org/authors/?q=ai:dangalchev.chavdar-atanasovSummary: The additional closeness is a very important characteristic of graphs. It measures the maximal closeness of a graph after adding a new link and it is an indication of the growth potential of graphs' closeness. Most of the time calculating the additional closeness requires solving nontrivial optimization problems. In this article, the additional closenesses of cycles, gear, and some other graphs are calculated. Bounds for additional closeness of graphs are discussed.Bumblebee visitation problemhttps://zbmath.org/1522.902412023-12-07T16:00:11.105023Z"Das, Sandip"https://zbmath.org/authors/?q=ai:das.sandip"Gahlawat, Harmender"https://zbmath.org/authors/?q=ai:gahlawat.harmenderSummary: Bumblebee visitation problem is defined on connected graphs where a mobile agent, called Bumblebee, moves along the edges under some rules to achieve some optimization function. We prove this problem to be NP-hard for general graphs. We present a linear time algorithm for this problem on trees.
For the entire collection see [Zbl 1410.68018].On modelling and solving the shortest path problem with evidential weightshttps://zbmath.org/1522.902422023-12-07T16:00:11.105023Z"Vu, Tuan-Anh"https://zbmath.org/authors/?q=ai:vu.tuan-anh"Afifi, Sohaib"https://zbmath.org/authors/?q=ai:afifi.sohaib"Lefèvre, Éric"https://zbmath.org/authors/?q=ai:lefevre.eric"Pichon, Frédéric"https://zbmath.org/authors/?q=ai:pichon.fredericSummary: We study the single source single destination shortest path problem in a graph where information about arc weights is modelled by a belief function. We consider three common criteria to compare paths with respect to their weights in this setting: generalized Hurwicz, strong dominance and weak dominance. We show that in the particular case where the focal sets of the belief function are Cartesian products of intervals, finding best, i.e., non-dominated, paths according to these criteria amounts to solving known variants of the deterministic shortest path problem, for which exact resolution algorithms exist.
For the entire collection see [Zbl 1511.68018].New LP relaxations for minimum cycle/path/tree cover problemshttps://zbmath.org/1522.902432023-12-07T16:00:11.105023Z"Yu, Wei"https://zbmath.org/authors/?q=ai:yu.wei"Liu, Zhaohui"https://zbmath.org/authors/?q=ai:liu.zhaohui"Bao, Xiaoguang"https://zbmath.org/authors/?q=ai:bao.xiaoguangSummary: Given an undirected complete weighted graph \(G=(V,E)\) with nonnegative weight function obeying the triangle inequality, a set \(\{C_1,C_2,\ldots ,C_k\}\) of cycles is called a \textit{cycle cover} if \(V \subseteq \bigcup_{i=1}^k V(C_i)\) and its cost is given by the maximum weight of the cycles. The Minimum Cycle Cover Problem (MCCP) aims to find a cycle cover of cost at most \(\lambda\) with the minimum number of cycles. We propose new LP relaxations for MCCP as well as its variants, called the Minimum Path Cover Problem (MPCP) and the Minimum Tree Cover Problem, where the cycles are replaced by paths or trees. Moreover, we give new LP relaxations for a special case of the rooted version of MCCP/MPCP and show that these LP relaxations have significantly better integrality gaps than the previous relaxations.
For the entire collection see [Zbl 1400.68037].A unified greedy approximation for several dominating set problemshttps://zbmath.org/1522.902442023-12-07T16:00:11.105023Z"Zhong, Hao"https://zbmath.org/authors/?q=ai:zhong.hao"Tang, Yong"https://zbmath.org/authors/?q=ai:tang.yong"Zhang, Qi"https://zbmath.org/authors/?q=ai:zhang.qi.2|zhang.qi.12|zhang.qi.8|zhang.qi.7|zhang.qi-shuhuason|zhang.qi.4"Lin, Ronghua"https://zbmath.org/authors/?q=ai:lin.ronghua"Li, Weisheng"https://zbmath.org/authors/?q=ai:li.weishengSummary: Minimum Dominating Set and Minimum Connected Dominating Set are classic graph problems that have been studied extensively in the literature. These two problems and their various variants are NP-hard in a general graph, and for some of them greedy approximation algorithms have been proposed. In this paper, by designing two potential functions that enjoy submodularity or a weak submodularity, we propose a unified \(O(\ln \delta)\)-approximation algorithm for a generalized Minimum (Connected) Dominating Set that includes Minimum (Connected) Dominating Set, Minimum (Connected) Total Dominating Set, Minimum (Connected) *-Dominating Set and Minimum (Connected) Positive Influence Dominating Set, where \(\delta\) is the maximum node degree of the input graph. For each specific version of the generalized Minimum (Connected) Dominating Set, the unified algorithm either matches the best one of existing approximation algorithms, or gives the first approximation solution.Dual ascent and primal-dual algorithms for infinite-horizon nonstationary Markov decision processeshttps://zbmath.org/1522.902452023-12-07T16:00:11.105023Z"Ghate, Archis"https://zbmath.org/authors/?q=ai:ghate.archisSummary: Infinite-horizon nonstationary Markov decision processes (MDPs) extend their stationary counterparts by allowing temporal variations in immediate costs and transition probabilities. Bellman's characterization of optimality and equivalent primal-dual linear programming formulations for these MDPs include a countably infinite number of variables and equations. Simple policy iteration, also viewed as a primal simplex algorithm, is the state of the art in solving these MDPs. It produces a sequence of policies whose costs-to-go converge monotonically from above to optimal. This suffers from two limitations. A cost-improving policy update is computationally expensive and an optimality gap is missing. We propose two dual-based approaches to address these concerns. The first, called dual ascent, maintains approximate costs-to-go (dual variables) and corresponding nonnegative errors in Bellman's equations. The dual variables are iteratively increased such that errors vanish asymptotically. This guarantees that dual variables converge monotonically from below to optimal. This has two limitations. It does not maintain a sequence of policies (primal variables). Hence, it does not provide a decision-making strategy at termination and does not offer an upper bound on the optimal costs-to-go. The second approach, termed the primal-dual method, addresses these limitations. It maintains a primal policy, dual approximations of its costs-to-go, the corresponding nonegative Bellman's errors, and inherits monotonic dual value convergence. The key is a so-called rebalancing step, which leads to a duality gap-based stopping criterion and also primal value convergence. Computational experiments demonstrate the benefits of primal-dual over dual ascent and that primal-dual is orders of magnitude faster than simple policy iteration.Average criteria in denumerable semi-Markov decision chains under risk-aversionhttps://zbmath.org/1522.902462023-12-07T16:00:11.105023Z"Cavazos-Cadena, Rolando"https://zbmath.org/authors/?q=ai:cavazos-cadena.rolando"Cruz-Suárez, Hugo"https://zbmath.org/authors/?q=ai:cruz-suarez.hugo-adan"Montes-De-Oca, Raúl"https://zbmath.org/authors/?q=ai:montes-de-oca.raulSummary: This note concerns with semi-Markov decision chains evolving on a denumerable state space. The system is directed by a risk-averse controller with constant risk-sensitivity, and the performance of a decision policy is measured by a long-run average criterion associated with bounded holding cost rates and one-step cost function. Under mild conditions on the sojourn times and the transition law, restrictions on the cost structure are given to ensure that the optimal average cost can be characterized via a bounded solution of the optimality equation. Such a result is used to establish a general characterization of the optimal average cost in terms of an optimality inequality from which an optimal stationary policy can be derived.Duality between large deviation control and risk-sensitive control for Markov decision processeshttps://zbmath.org/1522.902472023-12-07T16:00:11.105023Z"Dai, Yanan"https://zbmath.org/authors/?q=ai:dai.yanan"Chen, Jinwen"https://zbmath.org/authors/?q=ai:chen.jinwenSummary: This paper studies the dual relation between large deviations control of maximizing ``up-side chance'' probability and risk-sensitive control for Markov Decision Processes. To derive the desired duality, we apply a non-linear extension of the Kreĭn-Rutman Theorem to characterize the optimal risk-sensitive value and prove that an optimal policy exists which is stationary and deterministic. Benchmarks in the ``up-side chance'' probability which make the duality hold are characterized. It is proved that the optimal policy for the ``up-side chance'' probability can be approximated by the optimal one for the risk-sensitive control. The right-hand derivative of the optimal risk-sensitive value function plays an important role, and a variational formula for the optimal risk-sensitive value is applied to characterize it. Some essential differences between these two types of optimal control problems are presented.Interval dominance based structural results for Markov decision processhttps://zbmath.org/1522.902482023-12-07T16:00:11.105023Z"Krishnamurthy, Vikram"https://zbmath.org/authors/?q=ai:krishnamurthy.vikramSummary: Structural results impose sufficient conditions on the model parameters of a Markov decision process (MDP) so that the optimal policy is an increasing function of the underlying state. The classical assumptions for MDP structural results require supermodularity of the rewards and transition probabilities. However, supermodularity does not hold in many applications. This paper uses a sufficient condition for interval dominance (called \(\mathcal{I}\)) proposed in the micro-economics literature, to obtain structural results for MDPs under more general conditions. We present several MDP examples where supermodularity does not hold, yet \(\mathcal{I}\) holds, and so the optimal policy is monotone; these include sigmoidal rewards (arising in prospect theory for human decision making), bi-diagonal and perturbed bi-diagonal transition matrices (in optimal allocation problems). We also consider MDPs with TP3 transition matrices and concave value functions. Finally, reinforcement learning algorithms that exploit the differential sparse structure of the optimal monotone policy are discussed.Block policy mirror descenthttps://zbmath.org/1522.902492023-12-07T16:00:11.105023Z"Lan, Guanghui"https://zbmath.org/authors/?q=ai:lan.guanghui"Li, Yan"https://zbmath.org/authors/?q=ai:li.yan.24|li.yan.12|li.yan.21|li.yan.7|li.yan.43|li.yan.5|li.yan.15|li.yan.2|li.yan.25|li.yan.62|li.yan.41|li.yan.14|li.yan.9|li.yan|li.yan.11|li.yan.54|li.yan.28|li.yan.16|li.yan.10|li.yan.19"Zhao, Tuo"https://zbmath.org/authors/?q=ai:zhao.tuoSummary: In this paper, we present a new policy gradient (PG) method, namely, the block policy mirror descent (BPMD) method, for solving a class of regularized reinforcement learning (RL) problems with (strongly) convex regularizers. Compared to the traditional PG methods with a batch update rule, which visits and updates the policy for every state, the BPMD method has cheap per-iteration computation via a partial update rule that performs the policy update on a sampled state. Despite the nonconvex nature of the problem and a partial update rule, we provide a unified analysis for several sampling schemes and show that BPMD achieves fast linear convergence to the global optimality. In particular, uniform sampling leads to worst-case total computational complexity comparable to batch PG methods. A necessary and sufficient condition for convergence with on-policy sampling is also identified. With a hybrid sampling scheme, we further show that BPMD enjoys potential instance-dependent acceleration, leading to improved dependence on the state space and consequently outperforming batch PG methods. We then extend BPMD methods to the stochastic setting by utilizing stochastic first-order information constructed from samples. With a generative model, \(\widetilde{\mathcal{O}}(\vert{\mathcal{S}}\vert \vert{\mathcal{A}}\vert/\epsilon)\) (resp., \(\widetilde{\mathcal{O}}(\vert{\mathcal{S}}\vert \vert{\mathcal{A}}\vert/\epsilon^2))\) sample complexities are established for the strongly convex (resp., non-strongly convex) regularizers, where \(\epsilon\) denotes the target accuracy. To the best of our knowledge, this is the first time that block coordinate descent methods have been developed and analyzed for policy optimization in reinforcement learning, which provides a new perspective on solving large-scale RL problems.A note on the existence of optimal stationary policies for average Markov decision processes with countable stateshttps://zbmath.org/1522.902502023-12-07T16:00:11.105023Z"Xia, Li"https://zbmath.org/authors/?q=ai:xia.li"Guo, Xianping"https://zbmath.org/authors/?q=ai:guo.xianping"Cao, Xi-Ren"https://zbmath.org/authors/?q=ai:cao.xi-ren|cao.xirenSummary: In many practical stochastic dynamic optimization problems with countable states, the optimal policy possesses certain structural properties. For example, the \((s, S)\) policy in inventory control, the well-known \(c \mu\)-rule and the recently discovered \(c / \mu\)-rule [\textit{L. Xia} et al., ``A \(c/\mu\)-rule for job assignment in heterogeneous group-server queues'', Prod. Oper. Manag. 31, No. 3, 1191-1215 (2022; \url{doi:10.1111/poms.13605})] in scheduling of queues. A presumption of such results is that an optimal stationary policy exists. There are many research works regarding to the existence of optimal stationary policies of Markov decision processes with countable state spaces (see, e.g., \textit{D. P. Bertsekas} [Dynamic programming and optimal control. Vol. 2. Belmont, MA: Athena Scientific (2012; Zbl 1298.90001)]; \textit{O. Hernández-Lerma} and \textit{J. B. Lasserre} [Discrete-time Markov control processes. Basic optimality criteria. New York, NY: Springer-Verlag (1995; Zbl 0840.93001)]; \textit{M. L. Puterman} [Markov decision processes: discrete stochastic dynamic programming. New York, NY: John Wiley \& Sons, Inc. (1994; Zbl 0829.90134)]; \textit{L. I. Sennott} [Stochastic dynamic programming and the control of queueing systems. New York, NY: Wiley (1999; Zbl 0997.93503)]). However, these conditions are usually not easy to verify in such optimization problems. In this paper, we study the optimization of long-run average of continuous-time Markov decision processes with countable state spaces. We provide an intuitive approach to prove the existence of an optimal stationary policy. The approach is simply based on compactness of the policy space, with a special designed metric, and the continuity of the long-run average in the space. Our method is capable to handle cost functions unbounded from both above and below, which makes a complementary contribution to the literature work where the cost function is unbounded from only one side. Examples are provided to illustrate the application of our main results.First- and second-order optimality conditions for second-order cone and semidefinite programming under a constant rank conditionhttps://zbmath.org/1522.902512023-12-07T16:00:11.105023Z"Andreani, Roberto"https://zbmath.org/authors/?q=ai:andreani.roberto"Haeser, Gabriel"https://zbmath.org/authors/?q=ai:haeser.gabriel"Mito, Leonardo M."https://zbmath.org/authors/?q=ai:mito.leonardo-m"Ramírez, Héctor"https://zbmath.org/authors/?q=ai:ramirez.hector-c"Silveira, Thiago P."https://zbmath.org/authors/?q=ai:silveira.thiago-pSummary: The well known constant rank constraint qualification [Math. Program. Study 21, 110--126 (1984; Zbl 0549.90082)] introduced by \textit{R. Janin} for nonlinear programming has been recently extended to a conic context by exploiting the eigenvector structure of the problem. In this paper we propose a more general and geometric approach for defining a new extension of this condition to the conic context. The main advantage of our approach is that we are able to recast the strong second-order properties of the constant rank condition in a conic context. In particular, we obtain a second-order necessary optimality condition that is stronger than the classical one obtained under Robinson's constraint qualification, in the sense that it holds for every Lagrange multiplier, even though our condition is independent of Robinson's condition.A primal-dual finite element method for scalar and vectorial total variation minimizationhttps://zbmath.org/1522.902522023-12-07T16:00:11.105023Z"Hilb, Stephan"https://zbmath.org/authors/?q=ai:hilb.stephan"Langer, Andreas"https://zbmath.org/authors/?q=ai:langer.andreas|langer.andreas.1"Alkämper, Martin"https://zbmath.org/authors/?q=ai:alkamper.martinSummary: Based on the Fenchel duality we build a primal-dual framework for minimizing a general functional consisting of a combined \(L^1\) and \(L^2\) data-fidelity term and a scalar or vectorial total variation regularisation term. The minimization is performed over the space of functions of bounded variations and appropriate discrete subspaces. We analyze the existence and uniqueness of solutions of the respective minimization problems. For computing a numerical solution we derive a semi-smooth Newton method on finite element spaces and highlight applications in denoising, inpainting and optical flow estimation.Robust optimality conditions and duality for nonsmooth multiobjective fractional semi-infinite programming problems with uncertain datahttps://zbmath.org/1522.902532023-12-07T16:00:11.105023Z"Thuy, Nguyen Thi Thu"https://zbmath.org/authors/?q=ai:thuy.nguyen-thi-thu"Su, Tran Van"https://zbmath.org/authors/?q=ai:su.tran-vanSummary: In this article, some Karush-Kuhn-Tucker type robust optimality conditions and duality for an uncertain nonsmooth multiobjective fractional semi-infinite programming problem ((UMFP), for short) are established. First, we provide, by combining robust optimization and the robust limiting constraint qualification, robust necessary optimality conditions in terms of Mordukhovich's subdifferentials. Under suitable assumptions on the generalized convexity/the strictly generalized convexity, robust necessary optimality condition becomes robust sufficient optimality condition. Second, we formulate types of Mond-Weir and Wolfe robust dual problem for (UMFP) via the Mordukhovich subdifferentials. Finally, as an application, we establish weak/strong/converse robust duality theorems for the problem (UMFP) and its Mond-Weir and Wolfe types dual problem. Some illustrative examples are also provided for our findings.Robust nonsmooth optimality conditions for multiobjective optimization problems with infinitely many uncertain constraintshttps://zbmath.org/1522.902542023-12-07T16:00:11.105023Z"Wang, Jie"https://zbmath.org/authors/?q=ai:wang.jie.12"Li, Sheng-Jie"https://zbmath.org/authors/?q=ai:li.shengjie"Chen, Chun-Rong"https://zbmath.org/authors/?q=ai:chen.chunrongSummary: This paper concentrates on uncertain multiobjective optimization problems with an arbitrary number of uncertain constraints under nonconvex and nonsmooth assumptions. We analyse the properties of robust infinite constraints and arrive at the Clarke subdifferential of the double supremum function. Subsequently, we employ the obtained subdifferential rules to study KKT robust necessary and sufficient optimality conditions for two types of uncertain multiobjective optimization problems. Simultaneously, some illustrative examples are provided to show the validity of the main results. Our results are new and generalize several corresponding results in the literature.A note on ``Levitin-Polyak well-posedness in set optimization concerning Pareto efficiency''https://zbmath.org/1522.902552023-12-07T16:00:11.105023Z"Wang, Wenqing"https://zbmath.org/authors/?q=ai:wang.wenqing.2"Xu, Yihong"https://zbmath.org/authors/?q=ai:xu.yihongSummary: This note points out that Lemma 4.1 and Theorem 4.2 in [\textit{T. Q. Duy}, Positivity 25, No. 5, 1923--1942 (2021; Zbl 1481.90282)] fail to hold, we modify the Lemma 4.1 and Theorem 4.2.Solving strongly convex-concave composite saddle-point problems with low dimension of one group of variablehttps://zbmath.org/1522.902562023-12-07T16:00:11.105023Z"Alkousa, Mohammad S."https://zbmath.org/authors/?q=ai:alkousa.mohammad-s"Gasnikov, Alexander V."https://zbmath.org/authors/?q=ai:gasnikov.aleksandr-v"Gladin, Egor L."https://zbmath.org/authors/?q=ai:gladin.egor-l"Kuruzov, Ilya A."https://zbmath.org/authors/?q=ai:kuruzov.ilya-a"Pasechnyuk, Dmitry A."https://zbmath.org/authors/?q=ai:pasechnyuk.dmitry-a"Stonyakin, Fedor S."https://zbmath.org/authors/?q=ai:stonyakin.fedor-sergeevichSummary: Algorithmic methods are developed that guarantee efficient complexity estimates for strongly convex-concave saddle-point problems in the case when one group of variables has a high dimension, while another has a rather low dimension (up to 100). These methods are based on reducing problems of this type to the minimization (maximization) problem for a convex (concave) functional with respect to one of the variables such that an approximate value of the gradient at an arbitrary point can be obtained with the required accuracy using an auxiliary optimization subproblem with respect to the other variable. It is proposed to use the ellipsoid method and Vaidya's method for low-dimensional problems and accelerated gradient methods with inexact information about the gradient or subgradient for high-dimensional problems. In the case when one group of variables, ranging over a hypercube, has a very low dimension (up to five), another proposed approach to strongly convex-concave saddle-point problems is rather efficient. This approach is based on a new version of a multidimensional analogue of Nesterov's method on a square (the multidimensional dichotomy method) with the possibility to use inexact values of the gradient of the objective functional.Alternating proximal-gradient steps for (stochastic) nonconvex-concave minimax problemshttps://zbmath.org/1522.902572023-12-07T16:00:11.105023Z"Boţ, Radu Ioan"https://zbmath.org/authors/?q=ai:bot.radu-ioan"Böhm, Axel"https://zbmath.org/authors/?q=ai:bohm.axelSummary: Minimax problems of the form \(\min_x\max_y\Psi (x,y)\) have attracted increased interest largely due to advances in machine learning, in particular generative adversarial networks and adversarial learning. These are typically trained using variants of stochastic gradient descent for the two players. Although convex-concave problems are well understood with many efficient solution methods to choose from, theoretical guarantees outside of this setting are sometimes lacking even for the simplest algorithms. In particular, this is the case for alternating gradient descent ascent, where the two agents take turns updating their strategies. To partially close this gap in the literature we prove a novel global convergence rate for the stochastic version of this method for finding a critical point of \(\psi(\cdot):=\max_y\Psi(\cdot,y)\) in a setting which is not convex-concave.The landscape of the proximal point method for nonconvex-nonconcave minimax optimizationhttps://zbmath.org/1522.902582023-12-07T16:00:11.105023Z"Grimmer, Benjamin"https://zbmath.org/authors/?q=ai:grimmer.benjamin"Lu, Haihao"https://zbmath.org/authors/?q=ai:lu.haihao"Worah, Pratik"https://zbmath.org/authors/?q=ai:worah.pratik"Mirrokni, Vahab"https://zbmath.org/authors/?q=ai:mirrokni.vahab-sSummary: Minimax optimization has become a central tool in machine learning with applications in robust optimization, reinforcement learning, GANs, etc. These applications are often nonconvex-nonconcave, but the existing theory is unable to identify and deal with the fundamental difficulties this poses. In this paper, we study the classic proximal point method (PPM) applied to nonconvex-nonconcave minimax problems. We find that a classic generalization of the Moreau envelope by Attouch and Wets provides key insights. Critically, we show this envelope not only smooths the objective but can convexify and concavify it based on the level of interaction present between the minimizing and maximizing variables. From this, we identify three distinct regions of nonconvex-nonconcave problems. When interaction is sufficiently strong, we derive global linear convergence guarantees. Conversely when the interaction is fairly weak, we derive local linear convergence guarantees with a proper initialization. Between these two settings, we show that PPM may diverge or converge to a limit cycle.Optimality conditions for nonsmooth nonconvex-nonconcave min-max problems and generative adversarial networkshttps://zbmath.org/1522.902592023-12-07T16:00:11.105023Z"Jiang, Jie"https://zbmath.org/authors/?q=ai:jiang.jie.2|jiang.jie.4|jiang.jie.1|jiang.jie.6|jiang.jie.3|jiang.jie.5"Chen, Xiaojun"https://zbmath.org/authors/?q=ai:chen.xiaojun|chen.xiaojun.1Summary: This paper considers a class of nonsmooth nonconvex-nonconcave min-max problems in machine learning and games. We first provide sufficient conditions for the existence of global minimax points and local minimax points. Next, we establish the first-order and second-order optimality conditions for local minimax points by using directional derivatives. These conditions reduce to smooth min-max problems with Fréchet derivatives. We apply our theoretical results to generative adversarial networks (GANs) in which two neural networks contest with each other in a game. Examples are used to illustrate applications of the new theory for training GANs.Max-min problems of searching for two disjoint subsetshttps://zbmath.org/1522.902602023-12-07T16:00:11.105023Z"Khandeev, Vladimir"https://zbmath.org/authors/?q=ai:khandeev.vladimir-ilich"Neshchadim, Sergey"https://zbmath.org/authors/?q=ai:neshchadim.sergeySummary: The work considers three problems of searching for two disjoint subsets among a finite set of points in Euclidean space. In all three problems, it is required to maximize the minimal size of these subsets so that in each cluster, the total intra-cluster scatter of points relative to the cluster center does not exceed a predetermined threshold. In the first problem, the centers of the clusters are fixed points of Euclidean space and are given as input. In the second one, centers are unknown, but they belong to the initial set. In the last problem, the center of the cluster is the arithmetic mean of all its elements. Earlier works considered problems with constraints on the quadratic intra-cluster scatter.
Quadratic analogs of the first two problems were proven to be NP-hard even in the one-dimensional case. For the third analog, the complexity remains unknown. The main result of the work are proofs of NP-hardness of all considered problems even in the one-dimensional case.
For the entire collection see [Zbl 1508.90001].A unified single-loop alternating gradient projection algorithm for nonconvex-concave and convex-nonconcave minimax problemshttps://zbmath.org/1522.902612023-12-07T16:00:11.105023Z"Xu, Zi"https://zbmath.org/authors/?q=ai:xu.zi"Zhang, Huiling"https://zbmath.org/authors/?q=ai:zhang.huiling"Xu, Yang"https://zbmath.org/authors/?q=ai:xu.yang.1|xu.yang|xu.yang.3|xu.yang.2"Lan, Guanghui"https://zbmath.org/authors/?q=ai:lan.guanghuiSummary: Much recent research effort has been directed to the development of efficient algorithms for solving minimax problems with theoretical convergence guarantees due to the relevance of these problems to a few emergent applications. In this paper, we propose a unified single-loop alternating gradient projection (AGP) algorithm for solving smooth nonconvex-(strongly) concave and (strongly) convex-nonconcave minimax problems. AGP employs simple gradient projection steps for updating the primal and dual variables alternatively at each iteration. We show that it can find an \(\varepsilon\)-stationary point of the objective function in \({\mathcal{O}}\left(\varepsilon^{-2} \right)\) (resp. \({\mathcal{O}}\left(\varepsilon^{-4} \right))\) iterations under nonconvex-strongly concave (resp. nonconvex-concave) setting. Moreover, its gradient complexity to obtain an \(\varepsilon\)-stationary point of the objective function is bounded by \({\mathcal{O}}\left(\varepsilon^{-2} \right)\) (resp., \({\mathcal{O}}\left(\varepsilon^{-4} \right))\) under the strongly convex-nonconcave (resp., convex-nonconcave) setting. To the best of our knowledge, this is the first time that a simple and unified single-loop algorithm is developed for solving both nonconvex-(strongly) concave and (strongly) convex-nonconcave minimax problems. Moreover, the complexity results for solving the latter (strongly) convex-nonconcave minimax problems have never been obtained before in the literature. Numerical results show the efficiency of the proposed AGP algorithm. Furthermore, we extend the AGP algorithm by presenting a block alternating proximal gradient (BAPG) algorithm for solving more general multi-block nonsmooth nonconvex-(strongly) concave and (strongly) convex-nonconcave minimax problems. We can similarly establish the gradient complexity of the proposed algorithm under these four different settings.On tangent cone to systems of inequalities and equations in Banach spaces under relaxed constant rank conditionhttps://zbmath.org/1522.902622023-12-07T16:00:11.105023Z"Bednarczuk, E. M."https://zbmath.org/authors/?q=ai:bednarczuk.ewa-m"Leśniewski, K. W."https://zbmath.org/authors/?q=ai:lesniewski.k-w"Rutkowski, K. E."https://zbmath.org/authors/?q=ai:rutkowski.krzysztof-eSummary: Under the relaxed constant rank condition, introduced by \textit{L. Minchenko} and \textit{S. Stakhovski} [Optimization 60, No. 4--6, 429--440 (2011; Zbl 1250.90093)], we prove that the linearized cone is contained in the tangent cone (Abadie condition) for sets represented as solution sets to systems of finite numbers of inequalities and equations given by continuously differentiable functions defined on Banach spaces.Softmax policy gradient methods can take exponential time to convergehttps://zbmath.org/1522.902632023-12-07T16:00:11.105023Z"Li, Gen"https://zbmath.org/authors/?q=ai:li.gen.1"Wei, Yuting"https://zbmath.org/authors/?q=ai:wei.yuting"Chi, Yuejie"https://zbmath.org/authors/?q=ai:chi.yuejie"Chen, Yuxin"https://zbmath.org/authors/?q=ai:chen.yuxinSummary: The softmax policy gradient (PG) method, which performs gradient ascent under softmax policy parameterization, is arguably one of the de facto implementations of policy optimization in modern reinforcement learning. For \(\gamma\)-discounted infinite-horizon tabular Markov decision processes (MDPs), remarkable progress has recently been achieved towards establishing global convergence of softmax PG methods in finding a near-optimal policy. However, prior results fall short of delineating clear dependencies of convergence rates on salient parameters such as the cardinality of the state space \({\mathcal{S}}\) and the effective horizon \(\frac{1}{1-\gamma}\), both of which could be excessively large. In this paper, we deliver a pessimistic message regarding the iteration complexity of softmax PG methods, despite assuming access to exact gradient computation. Specifically, we demonstrate that the softmax PG method with stepsize \(\eta\) can take
\[
\frac{1}{\eta} |{\mathcal{S}}|^{2^{\Omega \left(\frac{1}{1-\gamma}\right)}} \text{ iterations}
\]
to converge, even in the presence of a benign policy initialization and an initial state distribution amenable to exploration (so that the distribution mismatch coefficient is not exceedingly large). This is accomplished by characterizing the algorithmic dynamics over a carefully-constructed MDP containing only three actions. Our exponential lower bound hints at the necessity of carefully adjusting update rules or enforcing proper regularization in accelerating PG methods.A restart scheme for the memoryless BFGS methodhttps://zbmath.org/1522.902642023-12-07T16:00:11.105023Z"Aminifard, Zohre"https://zbmath.org/authors/?q=ai:aminifard.zohre"Babaie-Kafaki, Saman"https://zbmath.org/authors/?q=ai:babaie-kafaki.samanSummary: Undesirable effects of the direction of the maximum magnification by the scaled memoryless BFGS (Broyden-Fletcher-Goldfarb-Shanno) updating formula are studied, including some probable computational errors as well as weakening the convergence. In light of the insight gained by our analyses, a restart criterion for the method is suggested to accelerate convergence and also to decrease the errors. Convergence analysis of the given method is carried out as well. At last, the efficiency of the proposed restart approach is investigated by numerical tests on a set of CUTEr problems.Regularization of limited memory quasi-Newton methods for large-scale nonconvex minimizationhttps://zbmath.org/1522.902652023-12-07T16:00:11.105023Z"Kanzow, Christian"https://zbmath.org/authors/?q=ai:kanzow.christian"Steck, Daniel"https://zbmath.org/authors/?q=ai:steck.daniel-danielSummary: This paper deals with regularized Newton methods, a flexible class of unconstrained optimization algorithms that is competitive with line search and trust region methods and potentially combines attractive elements of both. The particular focus is on combining regularization with limited memory quasi-Newton methods by exploiting the special structure of limited memory algorithms. Global convergence of regularization methods is shown under mild assumptions and the details of regularized limited memory quasi-Newton updates are discussed including their compact representations. Numerical results using all large-scale test problems from the CUTEst collection indicate that our regularized version of L-BFGS is competitive with state-of-the-art line search and trust-region L-BFGS algorithms and previous attempts at combining L-BFGS with regularization, while potentially outperforming some of them, especially when nonmonotonicity is involved.A family of the modified three-term Hestenes-Stiefel conjugate gradient method with sufficient descent and conjugacy conditionshttps://zbmath.org/1522.902662023-12-07T16:00:11.105023Z"Khoshsimaye-Bargard, Maryam"https://zbmath.org/authors/?q=ai:khoshsimaye-bargard.maryam"Ashrafi, Ali"https://zbmath.org/authors/?q=ai:ashrafi.ali-rezaSummary: To strengthen the three-term Hestenes-Stiefel conjugate gradient method proposed by Zhang et al., we suggest a modified version of it. For this purpose, by considering the Dai-Liao approach, the third term of Zhang et al. method is multiplied by a positive parameter which can be determined adaptively. To render an appropriate choice for the parameter of the search direction, we carry out a matrix analysis by which the sufficient descent property of the method is guaranteed. In the following, convergence analyses are discussed for convex and nonconvex cost functions. Eventually, numerical tests shed light on the efficiency of the performance of the proposed method.On implicit function theorem for locally Lipschitz equationshttps://zbmath.org/1522.902672023-12-07T16:00:11.105023Z"Arutyunov, Aram V."https://zbmath.org/authors/?q=ai:arutyunov.aram-v"Zhukovskiy, Sergey E."https://zbmath.org/authors/?q=ai:zhukovskiy.s-e|zhukovskiy.sergey-evgenyevichSummary: Equations defined by locally Lipschitz continuous mappings with a parameter are considered. Implicit function theorems for this equation are obtained. The regularity condition is formulated in the terms of the Clarke Jacobian. Implicit functions estimates are derived. It is shown that the considered regularity assumptions are weaker than most of the known ones. The obtained implicit function theorems are applied to derive conditions for upper semicontinuity of the optimal value function for parameterized optimization problemsNumerical study of applying complex-step gradient and Hessian approximations in derivative-free optimizationhttps://zbmath.org/1522.902682023-12-07T16:00:11.105023Z"Hare, Warren"https://zbmath.org/authors/?q=ai:hare.warren-l"Srivastava, Kashvi"https://zbmath.org/authors/?q=ai:srivastava.kashviSummary: We consider the problem of minimizing an objective function that is provided by an oracle. We assume that while the optimization problem seeks a real-valued solution, the oracle is capable of accepting complex-valued input and returning complex-valued output. We explore using complex-variables in order to approximate gradients and Hessians within a derivative-free optimization method. We provide several complex-variable based methods to construct approximate gradients and Hessians, then provide numerical error bounds for these methods. We apply the approximations in a Newton algorithm to numerically explore the pros and cons of each approximation technique.
Results find that complex-variable based methods improve the chances finding higher accuracy solutions and allow for smaller step sizes to be applied; but, require increased cpu time that outstrips any reduction in function calls used or iterations applied.Constrained optimization in the presence of noisehttps://zbmath.org/1522.902692023-12-07T16:00:11.105023Z"Oztoprak, Figen"https://zbmath.org/authors/?q=ai:oztoprak.figen"Byrd, Richard"https://zbmath.org/authors/?q=ai:byrd.richard-h"Nocedal, Jorge"https://zbmath.org/authors/?q=ai:nocedal.jorgeSummary: The problem of interest is the minimization of a nonlinear function subject to nonlinear equality constraints using a sequential quadratic programming (SQP) method. The minimization must be performed while observing only noisy evaluations of the objective and constraint functions. In order to obtain stability, the classical SQP method is modified by relaxing the standard Armijo line search based on the noise level in the functions, which is assumed to be known. Convergence theory is presented giving conditions under which the iterates converge to a neighborhood of the solution characterized by the noise level and the problem conditioning. The analysis assumes that the SQP algorithm does not require regularization or trust regions. Numerical experiments indicate that the relaxed line search improves the practical performance of the method on problems involving uniformly distributed noise, compared to a standard line search.Hyperparameter autotuning of programs with HybridTunerhttps://zbmath.org/1522.902702023-12-07T16:00:11.105023Z"Sauk, Benjamin"https://zbmath.org/authors/?q=ai:sauk.benjamin"Sahinidis, Nikolaos V."https://zbmath.org/authors/?q=ai:sahinidis.nikolaos-vSummary: Algorithms must often be tailored to a specific architecture and application in order to fully harness the capabilities of sophisticated computer architectures and computational implementations. However, the relationship between tuning parameters and performance is complicated and non-intuitive, having no explicit algebraic description. This is true particularly for programs such as GPU applications and compiler tuning, both of which have discrete and often nonlinear interactions between parameters and performance. After assessing a few alternative algorithmic configurations, we present two hybrid derivative-free optimization (DFO) approaches to maximize the performance of an algorithm. We demonstrate how we use our method to solve problems with up to 50 hyperparameters. When compared to state-of-the-art autotuners, our autotuner (a) reduces the execution time of dense matrix multiplication by a factor of 1.4x, (b) identifies high-quality tuning parameters in only \(5\%\) of the computational effort required by other autotuners, and (c) can be applied to any computer architecture. Our implementations of Bandit DFO and Hybrid DFO are publicly available at \url{https://github.com/bsauk/HybridTuner}.Multiple objective optimization with weighted superposition attraction-repulsion (moWSAR) algorithmhttps://zbmath.org/1522.902712023-12-07T16:00:11.105023Z"Baykasoğlu, Adil"https://zbmath.org/authors/?q=ai:baykasoglu.adilSummary: Weighted superposition attraction-repulsion (WSAR) algorithm is a recently proposed swarm intelligence-based metaheuristic algorithm that is inspired from superposition principle of physics and attracted movements of agents. WSAR has been applied to many single objective unconstrained and constrained complex optimization problems successfully. In the present study, WSAR is applied to multiple objective optimization (MOO) problems for the first time in the literature. Details of the WSAR algorithm along with applications to MOO problems that are collected from the literature are presented in this study. It is shown that WSAR is competitive and able to generate Pareto optimal solutions.
For the entire collection see [Zbl 1508.91005].Learning to optimizehttps://zbmath.org/1522.902722023-12-07T16:00:11.105023Z"Bremer, Jörg"https://zbmath.org/authors/?q=ai:bremer.jorgSummary: With a growing availability of ambient computing power as well as sensor data, networked systems are emerging in all areas of daily life. Coordination and optimization in complex cyber-physical systems demand for decentralized and self-organizing algorithms to cope with problem size and distributed information availability. Self-organization often relies on emergent behavior. Local observations and decisions aggregate to some global behavior without any apparent, explicitly programmed rule. Systematically designing algorithms with emergent behavior suitably for a new orchestration or optimization task is, at best, tedious and error prone. Appropriate design patterns are scarce so far. It is demonstrated that a machine learning approach based on Cartesian Genetic Programming is capable of learning the emergent mechanisms that are necessary for swarm-based optimization. Targeted emergent behavior can be evolved by evolution strategies. The learned swarm behavior is already significantly better than just random search. The encountered pitfalls as well as remaining challenges on the research agenda are discussed in detail. An additional fitness landscape analysis gives insight in obstructions during evolutions and clues for future improvements.
For the entire collection see [Zbl 1498.90005].Min-max-min optimization with smooth and strongly convex objectiveshttps://zbmath.org/1522.902732023-12-07T16:00:11.105023Z"Lamperski, Jourdain"https://zbmath.org/authors/?q=ai:lamperski.jourdain-b"Prokopyev, Oleg A."https://zbmath.org/authors/?q=ai:prokopyev.oleg-alexan"Wrabetz, Luca G."https://zbmath.org/authors/?q=ai:wrabetz.luca-gSummary: We consider min-max-min optimization with smooth and strongly convex objectives. Our motivation for studying this class of problems stems from its connection to the \(k\)-\textit{center problem} and the growing literature on \textit{min-max-min robust optimization.} In particular, the considered class of problems nontrivially generalizes the Euclidean \(k\)-center problem in the sense that \textit{distances} in this more general setting do not necessarily satisfy metric properties. We present a \(9\kappa\)-approximation algorithm (where \(\kappa\) is the maximum condition number of the functions involved) that generalizes a simple greedy 2-approximation algorithm for the classical \(k\)-center problem. We show that for any choice of \(\kappa\), there is an instance with a condition number of \(\kappa\) per which our algorithm yields a \((4\kappa+4\sqrt{\kappa}+1)\)-approximation guarantee, implying that our analysis is tight when \(\kappa=1\). Finally, we compare the computational performance of our approximation algorithm with an exact mixed integer linear programming approach.A projective splitting method for monotone inclusions: iteration-complexity and application to composite optimizationhttps://zbmath.org/1522.902742023-12-07T16:00:11.105023Z"Pentón Machado, Majela"https://zbmath.org/authors/?q=ai:penton-machado.majela"Sicre, Mauricio Romero"https://zbmath.org/authors/?q=ai:sicre.mauricio-romeroSummary: We propose an inexact projective splitting method to solve the problem of finding a zero of a sum of maximal monotone operators. We perform convergence and complexity analyses of the method by viewing it as a special instance of an inexact proximal point method proposed by Solodov and Svaiter in 2001, for which pointwise and ergodic complexity results have been studied recently by Sicre. Also, for this latter method, we establish convergence rates and complexity bounds for strongly monotone inclusions, from where we obtain linear convergence for our projective splitting method under strong monotonicity and cocoercivity assumptions. We apply the proposed projective splitting scheme to composite convex optimization problems and establish pointwise and ergodic function value convergence rates, extending a recent work of Johnstone and Eckstein.A quadratic optimization problem with bipolar fuzzy relation equation constraintshttps://zbmath.org/1522.902752023-12-07T16:00:11.105023Z"Abbasi, Molai A."https://zbmath.org/authors/?q=ai:abbasi.molai-ali(no abstract)Multi-attribute group decision-making for solid waste management using interval-valued \(q\)-rung orthopair fuzzy COPRAShttps://zbmath.org/1522.902762023-12-07T16:00:11.105023Z"Ahemad, Faizan"https://zbmath.org/authors/?q=ai:ahemad.faizan"Khan, Ahmad Zaman"https://zbmath.org/authors/?q=ai:khan.ahmad-zaman"Mehlawat, Mukesh Kumar"https://zbmath.org/authors/?q=ai:mehlawat.mukesh-kumar"Gupta, Pankaj"https://zbmath.org/authors/?q=ai:gupta.pankaj"Roy, Sankar Kumar"https://zbmath.org/authors/?q=ai:roy.sankar-kumarSummary: In this paper, the COPRAS (Complex Proportional Assessment) method is extended for interval-valued \(q\)-rung orthopair fuzzy numbers (IV \(q\)-ROFNs) to solve multi-attribute group decision-making (MAGDM) problems. A novel distance measure for IV \(q\)-ROFNs is proposed, and its properties are also probed. This distance measure is used in an improved weights determination method for decision-makers. A weighted projection optimization model is developed to evaluate the completely unknown attributes' weights. The projection of assessment values is defined by the positive and negative ideal solutions, which determine the resemblance between two objects by considering their directional angle. An Indian cities' ranking problem for a better solid waste management infrastructure is solved using the proposed approach based on composite indicators, like recycling waste, greenhouse gas emissions, waste generation, landfilling waste, recycling rate, waste-to-energy rate, and composting waste. Numerical comparisons, sensitivity analysis, and other relevant analyses are performed for validation.A new method to solve linear programming problems in the environment of picture fuzzy setshttps://zbmath.org/1522.902772023-12-07T16:00:11.105023Z"Akram, M."https://zbmath.org/authors/?q=ai:akram.muhammad"Ullah, I."https://zbmath.org/authors/?q=ai:ullah.inayat"Allahviranloo, T."https://zbmath.org/authors/?q=ai:allahviranloo.tofigh(no abstract)Minimizing a linear objective function under a max-overlap function fuzzy relational equation constrainthttps://zbmath.org/1522.902782023-12-07T16:00:11.105023Z"Fang, Bo Wen"https://zbmath.org/authors/?q=ai:fang.bo-wenSummary: Max-t-norm compositions are commonly utilized to optimize a linear objective function subject to fuzzy relational equations, especially, max-min and max-product compositions. However, the associativity is not forcefully needed in many cases. In this paper, the max-overlap function composition is considered for the same optimization model. Then some properties of the solution set are obtained. According to these properties, the characterization of the optimal solution for the optimization model is proposed. Furthermore, a simple value matrix with rules is proposed to reduce problem size. Thus, a solution procedure is presented for determining optimal solutions without translating such an optimization problem into two sub-problems. A numerical example is provided to illustrate the procedure.Hybrid ant colony optimization algorithms -- behaviour investigation based on intuitionistic fuzzy logichttps://zbmath.org/1522.902792023-12-07T16:00:11.105023Z"Fidanova, Stefka"https://zbmath.org/authors/?q=ai:fidanova.stefka"Ganzha, Maria"https://zbmath.org/authors/?q=ai:ganzha.maria"Roeva, Olympia"https://zbmath.org/authors/?q=ai:roeva.olympia-nSummary: The local search procedure is a method for hybridization and improvement of the main algorithm, when complex problems are solved. It helps to avoid local optimums and to find faster the global one. The theory of intuitionistic fuzzy logic, which is the basis of InterCriteria analysis (ICrA), is used to study the proposed hybrid algorithms for ant colony optimization (ACO). Different algorithms for ICrA implementation are applied on the results obtained by hybrid ACO algorithms for Multiple Knapsack Problem. The hybrid algorithms behavior is compared to the traditional ACO algorithm. Based on the obtained numerical results from the algorithms performance and from the ICrA the efficiency and effectiveness of the proposed hybrid ACO algorithms, combined with appropriate local search procedure, are confirmed.
For the entire collection see [Zbl 1498.90005].Distributionally robust possibilistic optimization problemshttps://zbmath.org/1522.902802023-12-07T16:00:11.105023Z"Guillaume, Romain"https://zbmath.org/authors/?q=ai:guillaume.romain"Kasperski, Adam"https://zbmath.org/authors/?q=ai:kasperski.adam"Zieliński, Paweł"https://zbmath.org/authors/?q=ai:zielinski.pawelSummary: In this paper a class of optimization problems with uncertain linear constraints is discussed. It is assumed that the constraint coefficients are random vectors whose probability distributions are only partially known. Possibility theory is used to model imprecise probabilities. In one of interpretation, a possibility distribution (a membership function of a fuzzy set) in the set of coefficient realizations induces a necessity measure, which in turn defines a family of probability distributions in this set. The distributionally robust approach is then used to transform the imprecise constraints into deterministic counterparts. Namely, the uncertain left-hand side of each constraint is replaced with the expected value with respect to the worst probability distribution that can occur. It is shown how to represent the resulting problem by using linear or second-order cone constraints. This leads to problems which are computationally tractable for a wide class of optimization models, in particular for linear programming.Further characterizations of possibility-theoretical indices in fuzzy optimizationhttps://zbmath.org/1522.902812023-12-07T16:00:11.105023Z"Ike, Koichiro"https://zbmath.org/authors/?q=ai:ike.koichiroSummary: In this paper, we investigate six types of fuzzy relations based on Zadeh's possibility theory. Such relations were originally proposed by \textit{D. Dubois} and \textit{H. Prade} [Inf. Sci. 30, 183--224 (1983; Zbl 0569.94031)] as comparison indices for fuzzy numbers and were extended by \textit{M. Inuiguchi} et al. [Inf. Sci. 61, No. 3, 187--209 (1992; Zbl 0763.90001); Fuzzy Sets Syst. 49, No. 3, 243--259 (1992; Zbl 0786.90090); Inf. Sci. 67, No. 1--2, 93--126 (1993; Zbl 0770.90078)] to general cases. The fuzzy relations we consider are defined for fuzzy sets in a vector space preordered by a convex cone. We reveal the connection of the fuzzy relations with set relations, order-like crisp relations known in the area of set optimization. Specifically, we prove some equalities and equivalences involving the fuzzy relations and set relations under three different assumptions. We finally consider a fuzzy optimization problem in a general form and define its solution concepts. Applying the above equivalences to the problem, we obtain its relationship to certain set optimization problems.Soft robust solutions to possibilistic optimization problemshttps://zbmath.org/1522.902822023-12-07T16:00:11.105023Z"Kasperski, Adam"https://zbmath.org/authors/?q=ai:kasperski.adam"Zieliński, Paweł"https://zbmath.org/authors/?q=ai:zielinski.pawelSummary: This paper discusses a class of uncertain optimization problems, in which unknown parameters are modeled by fuzzy intervals. The membership functions of the fuzzy intervals are interpreted as possibility distributions for the values of the uncertain parameters. It is shown how the known concepts of robustness and light robustness, for the traditional interval uncertainty representation of the parameters, can be generalized to choose solutions that optimize against plausible parameter realizations under the assumed model of uncertainty in the possibilistic setting. Furthermore, these solutions can be computed efficiently for a wide class of problems, in particular for linear programming problems with fuzzy parameters in constraints and objective function. Thus the problems under consideration are not much computationally harder than their deterministic counterparts. In this paper a theoretical framework is presented and results of some computational tests are shown.Optimization problems with random datahttps://zbmath.org/1522.902832023-12-07T16:00:11.105023Z"Popova, Olga A."https://zbmath.org/authors/?q=ai:popova.olga-arkadevnaSummary: The article discusses a new approach to optimization problems with random input parameters, which is defined as a random programming. This approach uses a numerical probability analysis and allows us to construct the set of solutions of the optimization problem based on the joint probability density function.Optimality conditions for fuzzy optimization in several variables under generalized differentiabilityhttps://zbmath.org/1522.902842023-12-07T16:00:11.105023Z"Qiu, Dong"https://zbmath.org/authors/?q=ai:qiu.dong"Ouyang, Chenxi"https://zbmath.org/authors/?q=ai:ouyang.chenxiSummary: After giving a calculation formula for the g-derivative for a fuzzy-valued function with unilateral derivatives of the endpoint functions of level sets and characterizing the partial order relations between two fuzzy intervals by their generalized difference, we present the optimality conditions of weak LU-minimum solutions for fuzzy optimization problems by using the partial generalized differentiability and directional generalized differentiability. The optimization conditions obtained in this article extend the results in the literature. Examples are also given to illustrate the main theorems.A novel interval linear programming based on probabilistic dominancehttps://zbmath.org/1522.902852023-12-07T16:00:11.105023Z"Qiu, Zhiping"https://zbmath.org/authors/?q=ai:qiu.zhiping"Xia, Haijun"https://zbmath.org/authors/?q=ai:xia.haijunSummary: This study investigates a novel interval linear programming based on probabilistic dominance. Firstly, the definition of interval linear programming is briefly reviewed. Then a new interval linear programming model is presented based on probabilistic dominance. The probabilistic dominance index treats the intervals as uniformly distributed variables and the interval inequality relation is further defined by probability. To deal with non-linearity in probabilistic dominance index, sequential quadratic programming is used to solve the problem and the performance measure approach is proposed to overcome the convergence difficulties. The determination and sensitivity analysis of the target performance measure are discussed to assess the sequential quadratic programming algorithm. Meanwhile, the extension of the proposed method to fuzzy interval linear programming is discussed. Furthermore, the proposed method is applied to the design of the plane truss structure with interval parameters. Finally, the effectiveness and rationality of the developed method are demonstrated by two mathematical examples and one interval parametric plane truss structure optimization example.Solving max-Archimedean \(t\)-norm interval-valued fuzzy relation equationshttps://zbmath.org/1522.902862023-12-07T16:00:11.105023Z"Tiwari, Vijay Lakshmi"https://zbmath.org/authors/?q=ai:tiwari.vijay-lakshmi"Thapar, Antika"https://zbmath.org/authors/?q=ai:thapar.antikaSummary: This paper discusses a new method for finding the complete set of tolerable solutions of max-Archimedean interval-valued fuzzy relation equations. According to the literature, three types of solution sets, namely; tolerable solution set, united solution set and controllable solution set can be identified with interval-valued fuzzy relation equations. The structure and the properties of the tolerable solution set are studied. The complete set of tolerable solutions can be characterized by one maximum solution and finitely many minimal solutions. An efficient method based on the concept of covering is proposed which computes all minimal solutions. The concept of covering is useful for large size problems in terms of computation. The proposed method is illustrated with some examples.On fuzzy linearization approaches for solving multi-objective linear fractional programming problemshttps://zbmath.org/1522.902872023-12-07T16:00:11.105023Z"Valipour, Ezat"https://zbmath.org/authors/?q=ai:valipour.ezat"Yaghoobi, Mohammad Ali"https://zbmath.org/authors/?q=ai:yaghoobi.mohammad-aliSummary: This study first surveys fuzzy linearization approaches for solving multi-objective linear fractional programming (MOLFP) problems. In particular, we review different existing methods dealing with fuzzy objectives on a crisp constraint set. Those methods transform the given MOLFP problem into a linear or a multi-objective linear programming (LP or MOLP) problem and obtain one efficient or weakly efficient solution of the main MOLFP problem. We show that one of these popular existing methods has shortcomings, and we modify it to be able to find efficient solutions. The main idea of LP-based methods is optimizing a weighted sum of numerators and the negative form of denominators of the given fractional objective function over the feasible set. We prove there is no weight region to guarantee the efficiency of the optimal solutions of such LP-based methods whenever the interior of the feasible set is nonempty. Moreover, MOLP-based methods obtain an equivalent MOLP problem to the main MOLFP problem using fuzzy set techniques. We prove MOLFP problems with a non-closed efficient set are not equivalent to MOLP ones whenever the equivalency mapping is continuous.Optimality conditions for interval valued optimization problemshttps://zbmath.org/1522.902882023-12-07T16:00:11.105023Z"Villanueva, Fabiola R."https://zbmath.org/authors/?q=ai:villanueva.fabiola-roxana"de Oliveira, Valeriano A."https://zbmath.org/authors/?q=ai:de-oliveira.valeriano-antunes"Costa, Tiago M."https://zbmath.org/authors/?q=ai:costa.tiago-mSummary: This work addresses constrained optimization problems in which the objective function is interval-valued while the inequality constraints functions are real-valued. Both necessary and sufficient optimality conditions are derived. They are given through the gH-gradient and the gH-directional derivative of the interval objective function. The necessary ones are of KKT-type. The sufficient conditions are of generalized convexity type. The developed theory is illustrated by means of some numerical examples.Generalized min-max programming problems subject to addition-min fuzzy relational inequalitieshttps://zbmath.org/1522.902892023-12-07T16:00:11.105023Z"Wu, Yan-Kuen"https://zbmath.org/authors/?q=ai:wu.yan-kuen"Chiu, Ya-Ling"https://zbmath.org/authors/?q=ai:chiu.ya-ling"Guu, Sy-Ming"https://zbmath.org/authors/?q=ai:guu.sy-ming|guu.symingSummary: In this paper, we explore a new \textit{generalized} min-max programming problem with constraints of addition-min fuzzy relational inequalities. This new generalized min-max programming model provides unified settings that enable the system manager to understand the system congestion level or the worst individual cost in the BitTorrent-like peer-to-peer file-sharing system. Theoretical results are presented to illustrate how the optimal value of generalized min-max programming problem can be obtained by solving a single-variable optimization model. Two approaches (an analytic method and an iterative approach) are provided to solve this single-variable optimization model. The complexity analyses of these two approaches are provided. Numerical examples demonstrate our proposed approaches.An active-set approach to finding a minimal-optimal solution to the min-max programming problem with addition-min fuzzy relational inequalitieshttps://zbmath.org/1522.902902023-12-07T16:00:11.105023Z"Wu, Yan-Kuen"https://zbmath.org/authors/?q=ai:wu.yan-kuen"Guu, Sy-Ming"https://zbmath.org/authors/?q=ai:guu.syming|guu.sy-mingSummary: In the literature, a BitTorrent-like peer-to-peer (BT-P2P) file-sharing system has been modeled as a system of fuzzy relational inequalities (FRI) with addition-min composition. And a min-max programming problem has been proposed to study its system congestion. From a cost-saving viewpoint, optimal solutions to the min-max programming problem may not be the minimal-optimal solution. The ``minimal'' solution gives better cost performance while the ``optimal'' solution gives the least system congestion. Such a drawback has been studied in the literature. In this paper, we propose a simple active-set approach to finding a cost-saving optimal solution, i.e. a minimal-optimal solution to the min-max programming problem. The complexity of our approach is \(O(m^2 n)\), where \(m\) is the number of decision variables and \(n\) is the number of constraints. Numerical examples are given to illustrate our procedures. Since our active-set approach depends on the given order of decision variables, by using a different sequence of decision variables, it may be able to find other minimal-optimal solutions. Therefore, our method may be able to provide more choices for the manager to use in decision-making.Analytical method for solving max-min inverse fuzzy relationhttps://zbmath.org/1522.902912023-12-07T16:00:11.105023Z"Wu, Yan-Kuen"https://zbmath.org/authors/?q=ai:wu.yan-kuen"Lur, Yung-Yih"https://zbmath.org/authors/?q=ai:lur.yung-yih"Wen, Ching-Feng"https://zbmath.org/authors/?q=ai:wen.chingfeng"Lee, Shie-Jue"https://zbmath.org/authors/?q=ai:lee.shie-jueSummary: This paper focuses on a classical problem of computing max-min inverse fuzzy relation. The resolution for the problem is useful for solving the well known problems of fuzzy abductive/backward reasoning. We propose a simple analytical method for finding its exact solutions or approximate solutions. Some examples are given to illustrate our results.Random-term-absent addition-min fuzzy relation inequalities and their lexicographic minimum solutionshttps://zbmath.org/1522.902922023-12-07T16:00:11.105023Z"Yang, Xiao-Peng"https://zbmath.org/authors/?q=ai:yang.xiaopengSummary: Addition-min fuzzy relation inequalities are a recently proposed model for describing the transmission mechanism in a peer-to-peer (P2P) network system. Considering the random line faults that take place between two terminals in a P2P network system, we propose a new kind of model, namely, random-term-absent (RTA) addition-min fuzzy relation inequalities. Some of their relevant properties are investigated in this paper. In addition, to decrease the network congestion under a fixed priority grade of the terminals, we introduce the lexicographic minimum solutions of RTA addition-min fuzzy relation inequalities. An efficient algorithm is developed to find the unique lexicographic minimum solution with an illustrative example.Optimality conditions for fuzzy optimization problems under granular convexity concepthttps://zbmath.org/1522.902932023-12-07T16:00:11.105023Z"Zhang, Jianke"https://zbmath.org/authors/?q=ai:zhang.jianke"Chen, Xiaoyi"https://zbmath.org/authors/?q=ai:chen.xiaoyi"Li, Lifeng"https://zbmath.org/authors/?q=ai:li.lifeng"Ma, Xiaojue"https://zbmath.org/authors/?q=ai:ma.xiaojueSummary: In this paper, under the condition of granular differentiation, we consider the fuzzy optimization problems with the general fuzzy function as the objective function. Firstly, we introduce the concept of granular convexity, and propose the properties of the granular convex fuzzy functions. Secondly, we present the Karush-Kuhn-Tucker type optimality conditions of the fuzzy relative optimal solution of more general fuzzy programming problems and some test examples. Finally, the relationships between a class of variational inequalities and the fuzzy optimization problems are established.Optimal resource allocation model in disaster situations for maximizing the value of operational process resiliency and continuityhttps://zbmath.org/1522.902942023-12-07T16:00:11.105023Z"Ebrahimi-Sadrabadi, Mahnaz"https://zbmath.org/authors/?q=ai:ebrahimi-sadrabadi.mahnaz"Ostadi, Bakhtiar"https://zbmath.org/authors/?q=ai:ostadi.bakhtiar"Sepehri, Mohammad Mehdi"https://zbmath.org/authors/?q=ai:sepehri.mohammad-mehdi"Kashan, Ali Husseinzadeh"https://zbmath.org/authors/?q=ai:husseinzadeh-kashan.aliSummary: Organizations need to apply resilience and business continuity in industry to protect themselves against the crises and destructive events. Also, the growing expansion of competition in the global market and the increasing crisis in the world have increased the importance of optimal resource allocation. With the optimal resource allocation, huge losses and damages to organizations are prevented. The problem of resource allocation can be raised alongside the criteria of process resilience and continuity. Therefore, organizations change their perspective from focusing solely on reducing vulnerability to increasing resilience and continuity against to accidents in crises and destructive situations. The objective of this paper is proposed a mathematical model for optimal resource allocation with the aim of minimizing the lack of process resilience and maximizing the process continuity. So, the organization can continue to operate with available resources in times of crisis and lack of resources. Also, main activities and processes are not interrupted by crises and destructive events. After solving the model using a case study (textile industry), the results of the model were described and it was found that destructive events were recovered before the tolerance threshold and crisis and destructive events did not interrupt activities and processes.Online prediction with history-dependent experts: the general casehttps://zbmath.org/1522.910122023-12-07T16:00:11.105023Z"Drenska, Nadejda"https://zbmath.org/authors/?q=ai:drenska.nadejda"Calder, Jeff"https://zbmath.org/authors/?q=ai:calder.jeffSummary: We study the problem of \textit{prediction of binary sequences} with expert advice in the online setting, which is a classic example of online machine learning. We interpret the binary sequence as the price history of a stock, and view the predictor as an investor, which converts the problem into a \textit{stock prediction problem}. In this framework, an investor, who predicts the daily movements of a stock, and an adversarial market, who controls the stock, play against each other over turns. The investor combines the predictions of \(n \geq 2\) experts in order to make a decision about how much to invest at each turn, and aims to minimize their regret with respect to the best-performing expert at the end of the game. We consider the problem with \textit{history-dependent} experts, in which each expert uses the previous days of history of the market in making their predictions. We prove that the value function for this game, rescaled appropriately, converges as \(N \to \infty\) at a rate of \(O \left(N^{- 1 / 6}\right)\) to the viscosity solution of a nonlinear degenerate elliptic PDE, which can be understood as the Hamilton-Jacobi-Isaacs equation for the two-person game. As a result, we are able to deduce asymptotically optimal strategies for the investor. Our results extend those established by the first author and R.V. Kohn [14] for \(n = 2\) experts and \(d \leq 4\) days of history.
{{\copyright} 2022 The Authors. \textit{Communications on Pure and Applied Mathematics} published by Wiley Periodicals LLC.}A note on linear complementarity via two-person zero-sum gameshttps://zbmath.org/1522.910132023-12-07T16:00:11.105023Z"Dubey, Dipti"https://zbmath.org/authors/?q=ai:dubey.dipti"Neogy, S. K."https://zbmath.org/authors/?q=ai:neogy.samir-kumar"Raghavan, T. E. S."https://zbmath.org/authors/?q=ai:raghavan.tirukkannamangai-e-sSummary: The matrix \(M\) of a linear complementarity problem can be viewed as a payoff matrix of a two-person zero-sum game. Lemke's algorithm can be successfully applied to reach a complementary solution or infeasibility when the game satisfies the following conditions: (i) Value of \(M\) is equal to zero. (ii) For all principal minors of \(M^T\) (transpose of \(M)\) value is non-negative. (iii) For any optimal mixed strategy \(y\) of the maximizer either \(y_i>0\) or \((My)_i>0\) for each coordinate \(i\).Homotopy continuation method for discounted zero-sum stochastic game with ARAT structurehttps://zbmath.org/1522.910142023-12-07T16:00:11.105023Z"Dutta, A."https://zbmath.org/authors/?q=ai:dutta.animesh|dutta.abhraneel|dutta.amartya-kumar|dutta.anushree|dutta.anjan|dutta.aritra|dutta.arijit|dutta.amar-jyoti|dutta.aindrik|dutta.annwesha|dutta.amar-joyti|dutta.anamika|dutta.arpan|dutta.abhishek|dutta.aparna|dutta.arghya|dutta.anirban|dutta.anindita|dutta.anupam|dutta.abhijit|dutta.arun|dutta.avijit|dutta.ashin|dutta.ankan|dutta.arindam|dutta.ashit-kumar|dutta.ajoy|dutta.ayan|dutta.amitava|dutta.arnab|dutta.amit-kumar|dutta.argho|dutta.atri"Das, A. K."https://zbmath.org/authors/?q=ai:das.amit-kumar|das.ananga-kumar|kumar-das.ashish|das.amit-kumar.1|das.ajoy-k-r|das.amar-k|das.ajoy-kanti|das.arun-kumar|das.arup-kumar.1|das.ashok-kumar|das.anjan-kr|das.arup-kumar|das.asim-kumar|das.adway-kumar|das.abhik-kumar|das.asit-kumar|das.ashok-kumar.1|das.amal-k|das.arun-kumar.1|das.ashok-kSummary: In this paper, we introduce a new function to trace the trajectory by applying the modified homotopy continuation method for finding the solution of two-person zero-sum discounted stochastic ARAT game. For the proposed algorithm, the homotopy path approaching the solution is smooth and bounded.Correction to: ``On the polyhedral homotopy method for solving generalized Nash equilibrium problems of polynomials''https://zbmath.org/1522.910262023-12-07T16:00:11.105023Z"Lee, Kisun"https://zbmath.org/authors/?q=ai:lee.kisun"Tang, Xindong"https://zbmath.org/authors/?q=ai:tang.xindongCorrection to the authors' paper [ibid. 95, No. 1, Paper No. 13, 26 p. (2023; Zbl 1519.91013)].Partially observed discrete-time risk-sensitive mean field gameshttps://zbmath.org/1522.910422023-12-07T16:00:11.105023Z"Saldi, Naci"https://zbmath.org/authors/?q=ai:saldi.naci"Başar, Tamer"https://zbmath.org/authors/?q=ai:basar.tamer"Raginsky, Maxim"https://zbmath.org/authors/?q=ai:raginsky.maximSummary: In this paper, we consider discrete-time partially observed mean-field games with the risk-sensitive optimality criterion. We introduce risk-sensitivity behavior for each agent via an exponential utility function. In the game model, each agent is weakly coupled with the rest of the population through its individual cost and state dynamics via the empirical distribution of states. We establish the mean-field equilibrium in the infinite-population limit using the technique of converting the underlying original partially observed stochastic control problem to a fully observed one on the belief space and the dynamic programming principle. Then, we show that the mean-field equilibrium policy, when adopted by each agent, forms an approximate Nash equilibrium for games with sufficiently many agents. We first consider finite-horizon cost function and then discuss extension of the result to infinite-horizon cost in the next-to-last section of the paper.Integer-programming bounds on pebbling numbers of Cartesian-product graphshttps://zbmath.org/1522.910592023-12-07T16:00:11.105023Z"Kenter, Franklin"https://zbmath.org/authors/?q=ai:kenter.franklin-h-j"Skipper, Daphne"https://zbmath.org/authors/?q=ai:skipper.daphne-eSummary: Graph pebbling, as introduced by Chung, is a two-player game on a graph \(G\). Player one distributes ``pebbles'' to vertices and designates a root vertex. Player two attempts to move a pebble to the root vertex via a sequence of pebbling moves, in which two pebbles are removed from one vertex in order to place a single pebble on an adjacent vertex. The pebbling number of a simple graph \(G\) is the smallest number \(\pi _G\) such that if player one distributes \(\pi _G\) pebbles in \textit{any} configuration, player two can always win. Computing \(\pi _G\) is provably difficult, and recent methods for bounding \(\pi _G\) have proved computationally intractable, even for moderately sized graphs.
Graham conjectured that the pebbling number of the Cartesian-product of two graphs \(G\) and \(H\), denoted \(G\,\square \,H\), is no greater than \(\pi _G \pi _H\). Graham's conjecture has been verified for specific families of graphs; however, in general, the problem remains open.
This study combines the focus of developing a computationally tractable method for generating good bounds on \(\pi _{G \,\square \, H}\), with the goal of providing evidence for (or disproving) Graham's conjecture. In particular, we present a novel integer-programming (IP) approach to bounding \(\pi _{G \,\square \, H}\) that results in significantly smaller problem instances compared with existing IP approaches to graph pebbling. Our approach leads to a sizable improvement on the best known bound for \(\pi _{L \,\square \, L}\), where \(L\) is the Lemke graph. \(L\,\square \, L\) is among the smallest known potential counterexamples to Graham's conjecture.
For the entire collection see [Zbl 1407.68037].Dynamical Markov decision-making model based on mass function to quantitatively predict interference effectshttps://zbmath.org/1522.910852023-12-07T16:00:11.105023Z"Pan, Lipeng"https://zbmath.org/authors/?q=ai:pan.lipeng"Deng, Yong"https://zbmath.org/authors/?q=ai:deng.yong"Cheong, Kang Hao"https://zbmath.org/authors/?q=ai:cheong.kang-haoSummary: Experimental results demonstrate that the law of total probability which is used to manage probabilities of a number of decision stages, is violated when interference effects occur in the decision process. Although some attempts have been made to predict interference effects, these studies have only been able to do so for certain data while failing to do so for others in the same experiment. With the help of C-D experiment and D experiment, this paper develops a dynamical Markov decision-making model based on mass function to quantitatively predict the interference effects. This model employs both the mass function and discount coefficient to generate distribution of initial state. A transition matrix based on the characteristics of unitary matrix is then generated, which is capable of realizing both transition between adjacent states as well as constraining variation interval of the discount coefficient. Next, this model quantifies the difference between the two experimental results obtained through a probability transformation to predict interference effects. Finally, our proposed model is applied to existing dataset with the results indicating that our model can process all existing data associated with the experiments, as compared to other models.Application of an interval-valued intuitionistic fuzzy decision-making method in outsourcing using a software programhttps://zbmath.org/1522.910882023-12-07T16:00:11.105023Z"Traneva, Velichka"https://zbmath.org/authors/?q=ai:traneva.velichka"Tranev, Stoyan"https://zbmath.org/authors/?q=ai:tranev.stoyan"Mavrov, Deyan"https://zbmath.org/authors/?q=ai:mavrov.deyanSummary: Selecting a suitable candidate for outsourcing service provider is a challenging problem that requires discussion among a group of experts and the consideration of multiple criteria. The problem of this type belongs to the area of multicriteria decision-making. The imprecision in this problem may arise from the nature of the characteristics of the candidates for the service providers, which can be unavailable or indeterminate. It may also be derived from the inability of the experts to formulate a precise evaluation. Interval-valued intuitionistic fuzzy sets (IVIFSs), which are an extension of fuzzy sets, are the stronger tool in modeling uncertain problems than fuzzy ones. In this paper, which is an extension of \textit{V. Traneva} et al. [Stud. Comput. Intell. 1044, 215--232 (2022; Zbl 07720693)], we will further outline the principles of a software program for automated solution of an optimal interval-valued intuitionistic fuzzy multicriteria decision-making problem (IVIFIMOA) in outsourcing for the selection of the most appropriate candidates. As an example of a case study, an application of the algorithm on example company data is demonstrated.
For the entire collection see [Zbl 1498.90005].By bid-ask spreads towards competitive equilibriumhttps://zbmath.org/1522.911262023-12-07T16:00:11.105023Z"Flåm, Sjur Didrik"https://zbmath.org/authors/?q=ai:flam.sjur-didrikSummary: Consider agents who just use markets and money to mediate trade and express economic interests. Suppose \textit{bid-ask spreads} -- derived from generalized differentials -- derive and incite transactions. On that basis, this paper indicates good prospects for convergence towards \textit{competitive equilibrium}.
Presuming private property, main arguments hinge on three hooks: First, \textit{Fenchel conjugation} to record profits or values added; second, \textit{aggregate convexity} to support Pareto efficiency by prices; and third, \textit{absence of externalities} to make total additions of value dwindle step by step.A unified framework for pricing in nonconvex resource allocation gameshttps://zbmath.org/1522.911412023-12-07T16:00:11.105023Z"Harks, Tobias"https://zbmath.org/authors/?q=ai:harks.tobias"Schwarz, Julian"https://zbmath.org/authors/?q=ai:schwarz.julianSummary: We consider a basic nonconvex resource allocation game, where the players' strategy spaces are subsets of \(\mathbb{R}^m\) and cost functions are parameterized by some common vector \(u\in \mathbb{R}^m\) and, otherwise, only depend on their own strategy choice. A strategy of a player can be interpreted as a vector of resource consumption and a joint strategy profile naturally leads to an aggregate consumption vector. Resources can be priced, that is, the game is augmented by a price vector \(\lambda\in\mathbb{R}_{\geq 0}^m\) and players have quasi-linear overall costs, meaning that in addition to the original costs, a player needs to pay the corresponding price per consumed unit. We investigate the following question: for which aggregated consumption vectors \(u\) can we find prices \(\lambda\) that induce an equilibrium realizing the targeted consumption profile? For answering this question, we revisit a duality-based framework and derive a new characterization of the existence of such \(u\) and \(\lambda\) using convexification techniques. Our characterization implies the following result: If strategy spaces of players are bounded linear mixed-integer sets and the cost functions are linear or even concave, the equilibrium existence problem reduces to solving a well-structured LP. We then consider aggregate formulations assuming that cost functions are additive over resources and homogeneous among players. We derive a characterization of enforceable consumption vectors \(u\), showing that \(u\) is enforceable if and only if \(u\) is a minimizer of a certain convex optimization problem with a linear functional. We demonstrate that this framework can unify parts of four largely independent streams in the literature: tolls in transportation systems, Walrasian equilibria, trading networks, and congestion control. Besides reproving existing results we establish new enforceability results for these domains as well.Production network centrality in connection to economic development by the case of Kazakhstan statisticshttps://zbmath.org/1522.911432023-12-07T16:00:11.105023Z"Boranbayev, Seilkhan"https://zbmath.org/authors/?q=ai:boranbayev.seilkhan"Obrosova, Nataliia"https://zbmath.org/authors/?q=ai:obrosova.nataliia"Shananin, Alexander"https://zbmath.org/authors/?q=ai:shananin.aleksandr-aSummary: Analysis of a production network graph allows to determine the central industries of an economy. According to the well-known concept of economic development, these industries are the main drivers of economic growth. We discuss this concept in terms of a tractable model. Our approach is based on the nonlinear input-output balance model that is developed by authors. The classical method for determining of the drivers of economic growth is the Leontief's input-output model that has been widely used since the middle of the XX century. This model assumes the fixed proportions of material costs for a unit of product output. The nonlinear model is based on the more relevant assumption about the stability of the structure of financial costs of a production. We use a Cobb-Douglas production function in order to develop a technology that allows to analyze the nonlinear input-output balance. The technology is based on the solution of Fenchel duality problem of resource allocation. On the base of the obtained results we analyze the concept of centrality and the stability of intersectoral linkages by the case of Kazakhstan statistics.
For the entire collection see [Zbl 1508.90001].A lattice linear predicate parallel algorithm for the housing market problemhttps://zbmath.org/1522.911672023-12-07T16:00:11.105023Z"Garg, Vijay K."https://zbmath.org/authors/?q=ai:garg.vijay-k.1|garg.vijay-kSummary: It has been shown that lattice linear predicate (LLP) algorithm solves many combinatorial optimization problems such as the shortest path problem, the stable marriage problem and the market clearing price problem. In this paper, we give an LLP algorithm for the housing market problem. The housing market problem is a one-sided matching problem with \(n\) agents and \(n\) houses. Each agent has an initial allocation of a house and a totally ordered preference list of houses. The goal is to find a matching between agents and houses such that no strict subset of agents can improve their outcome by exchanging houses with each other rather than going with the matching. Gale's celebrated top trading cycle algorithm to find the matching requires \(O(n^2)\) time. Our parallel algorithm has expected time complexity \(O(n \log^2 n)\) with and expected work complexity of \(O(n^2 \log n)\).
For the entire collection see [Zbl 1509.68014].Influence maximization with latency requirements on social networkshttps://zbmath.org/1522.911932023-12-07T16:00:11.105023Z"Raghavan, S."https://zbmath.org/authors/?q=ai:raghavan.sekhar|raghavan.s-srinivasa|raghavan.sreenivasa-a|raghavan.srinivasacharya|raghavan.s-raghu|raghavan.srinivasan|raghavan.srikanth|raghavan.shuba-v"Zhang, Rui"https://zbmath.org/authors/?q=ai:zhang.rui.20Summary: Targeted marketing strategies are of significant interest in the smartapp economy. Typically, one seeks to identify individuals to strategically target in a social network so that the network is influenced at a minimal cost. In many practical settings, the effects of direct influence predominate, leading to the positive influence dominating set with partial payments (PIDS-PP) problem that we discuss in this paper. The PIDS-PP problem is NP-complete because it generalizes the dominating set problem. We discuss several mixed integer programming formulations for the PIDS-PP problem. First, we describe two compact formulations on the payment space. We then develop a stronger compact extended formulation. We show that when the underlying graph is a tree, this compact extended formulation provides integral solutions for the node selection variables. In conjunction, we describe a polynomial-time dynamic programming algorithm for the PIDS-PP problem on trees. We project the compact extended formulation onto the payment space, providing an equivalently strong formulation that has exponentially many constraints. We present a polynomial time algorithm to solve the associated separation problem. Our computational experience on a test bed of 100 real-world graph instances (with up to approximately 465,000 nodes and 835,000 edges) demonstrates the efficacy of our strongest payment space formulation. It finds solutions that are on average 0.4\% from optimality and solves 80 of the 100 instances to optimality.
Summary of contribution: The study of influence propagation is important in a number of applications including marketing, epidemiology, and healthcare. Typically, in these problems, one seeks to identify individuals to strategically target in a social network so that the entire network is influenced at a minimal cost. With the ease of tracking consumers in the smartapp economy, the scope and nature of these problems have become larger. Consequently, there is considerable interest across multiple research communities in computationally solving large-scale influence maximization problems, which thus represent significant opportunities for the development of operations research-based methods and analysis in this interface. This paper introduces the positive influence dominating set with partial payments (PIDS-PP) problem, an influence maximization problem where the effects of direct influence predominate, and it is possible to make partial payments to nodes that are not targeted. The paper focuses on model development to solve large-scale PIDS-PP problems. To this end, starting from an initial base optimization model, it uses several operations research model strengthening techniques to develop two equivalent models that have strong computational performance (and can be theoretically shown to be the best model for trees). Computational experiments on a test bed of 100 real-world graph instances (with up to approximately 465,000 nodes and 835,000 edges) attest to the efficacy of the best model, which finds solutions that are on average 0.4\% from optimality and solves 80 of the 100 instances to optimality.Rapid influence maximization on social networks: the positive influence dominating set problemhttps://zbmath.org/1522.911942023-12-07T16:00:11.105023Z"Raghavan, S."https://zbmath.org/authors/?q=ai:raghavan.s-raghu|raghavan.s-srinivasa|raghavan.sekhar|raghavan.srikanth|raghavan.srinivasacharya|raghavan.shuba-v|raghavan.srinivasan|raghavan.sreenivasa-a"Zhang, Rui"https://zbmath.org/authors/?q=ai:zhang.rui.20Summary: Motivated by applications arising on social networks, we study a generalization of the celebrated dominating set problem called the Positive Influence Dominating Set (PIDS). Given a graph \(G\) with a set \(V\) of nodes and a set \(E\) of edges, each node \(i\) in \(V\) has a weight \(b_i\), and a threshold requirement \(g_i\). We seek a minimum weight subset \(T\) of \(V\), so that every node \(i\) not in \(T\) is adjacent to at least \(g_i\) members of \(T\). When \(g_i\) is one for all nodes, we obtain the weighted dominating set problem. First, we propose a strong and compact extended formulation for the PIDS problem. We then project the extended formulation onto the space of the natural node-selection variables to obtain an equivalent formulation with an exponential number of valid inequalities. Restricting our attention to trees, we show that the extended formulation is the strongest possible formulation, and its projection (onto the space of the node variables) gives a complete description of the PIDS polytope on trees. We derive the necessary and sufficient facet-dening conditions for the valid inequalities in the projection and discuss their polynomial time separation. We embed this (exponential size) formulation in a branch-and-cut framework and conduct computational experiments using real-world graph instances, with up to approximately 2.5 million nodes and 8 million edges. On a test-bed of 100 real-world graph instances, our approach finds solutions that are on average 0.2\% from optimality and solves 51 out of the 100 instances to optimality.
Summary of contribution: In influence maximization problems, a decision maker wants to target individuals strategically to cause a cascade at a minimum cost over a social network. These problems have attracted significant attention as their applications can be found in many different domains including epidemiology, healthcare, marketing, and politics. However, computationally solving large-scale influence maximization problems to near optimality remains a substantial challenge for the computing community, which thus represent significant opportunities for the development of operations-research based models, algorithms, and analysis in this interface. This paper studies the positive influence dominating set (PIDS) problem, an influence maximization problem on social networks that generalizes the celebrated dominating set problem. It focuses on developing exact methods for solving large instances to near optimality. In other words, the approach results in strong bounds, which then provide meaningful comparative benchmarks for heuristic approaches. The paper first shows that straightforward generalizations of well-known formulations for the dominating set problem do not yield strong (i.e., computationally viable) formulations for the PIDS problem. It then strengthens these formulations by proposing a compact extended formulation and derives its projection onto the space on the natural node-selection variables, resulting in two equivalent (stronger) formulations for the PIDS problem. The projected formulation on the natural node-variables contains a new class of valid inequalities that are shown to be facet-defining for the PIDS problem. These theoretical results are complemented by in-depth computational experiments using a branch-and-cut framework, on a testbed of 100 real-world graph instances, with up to approximately 2.5 million nodes and 8 million edges. They demonstrate the effectiveness of the proposed formulation in solving large scale problems finding solutions that are on average 0.2\% from optimality and solving 51 of the 100 instances to optimality.Mean-variance-VaR portfolios: MIQP formulation and performance analysishttps://zbmath.org/1522.912112023-12-07T16:00:11.105023Z"Cesarone, Francesco"https://zbmath.org/authors/?q=ai:cesarone.francesco"Martino, Manuel L."https://zbmath.org/authors/?q=ai:martino.manuel-l"Tardella, Fabio"https://zbmath.org/authors/?q=ai:tardella.fabioSummary: Value-at-risk is one of the most popular risk management tools in the financial industry. Over the past 20 years, several attempts to include VaR in the portfolio selection process have been proposed. However, using VaR as a risk measure in portfolio optimization models leads to problems that are computationally hard to solve. In view of this, few practical applications of VaR in portfolio selection have appeared in the literature up to now. In this paper, we propose to add the VaR criterion to the classical mean-variance approach in order to better address the typical regulatory constraints of the financial industry. We thus obtain a portfolio selection model characterized by three criteria: expected return, variance, and VaR at a specified confidence level. The resulting optimization problem consists in minimizing variance with parametric constraints on the levels of expected return and VaR. This model can be formulated as a mixed-integer quadratic programming (MIQP) problem. An extensive empirical analysis on seven real-world datasets demonstrates the practical applicability of the proposed approach. Furthermore, the out-of-sample performance of the more binding optimal mean-variance-VaR portfolios seems to be generally better than that of the equally weighted and of the mean-variance-CVaR portfolios.Large-scale financial planning via a partially observable stochastic dual dynamic programming frameworkhttps://zbmath.org/1522.912292023-12-07T16:00:11.105023Z"Lee, Jinkyu"https://zbmath.org/authors/?q=ai:lee.jinkyu"Kwon, Do-Gyun"https://zbmath.org/authors/?q=ai:kwon.do-gyun"Lee, Yongjae"https://zbmath.org/authors/?q=ai:lee.yongjae"Kim, Jang Ho"https://zbmath.org/authors/?q=ai:kim.jangho|kim.jang-ho-robert"Kim, Woo Chang"https://zbmath.org/authors/?q=ai:kim.woo-changSummary: The multi-stage stochastic programming (MSP) approach is widely used to solve financial planning problems owing to its flexibility. However, the size of an MSP problem grows exponentially with the number of stages, and such problem can easily become computationally intractable. Financial planning problems often consider planning horizons of several decades, and thus, the curse of dimensionality can become a critical issue. Stochastic dual dynamic programming (SDDP), a sampling-based decomposition algorithm, has emerged to resolve this issue. While SDDP has been successfully implemented in the energy domain, few applications of SDDP are found in the finance domain. In this study, we identify the major obstacle in using SDDP to solve financial planning problems to be the stagewise independence assumption and propose a partially observable SDDP (PO-SDDP) framework to overcome such limitations. We argue that the PO-SDDP framework, which models uncertainties using discrete-valued partially observable Markov states and introduces feasibility cuts, can properly address large-scale financial planning problems.Co-jumps and recursive preferences in portfolio choiceshttps://zbmath.org/1522.912322023-12-07T16:00:11.105023Z"Oliva, Immacolata"https://zbmath.org/authors/?q=ai:oliva.immacolata"Stefani, Ilaria"https://zbmath.org/authors/?q=ai:stefani.ilariaSummary: This paper investigates a multivariate, dynamic, continuous-time optimal consumption and portfolio allocation problem when the investor faces recursive utilities. The economy we are considering is described through both diffusion and discontinuities in the dynamics. We derive an approximated closed-form solution to optimal rules by exploiting standard dynamic programming techniques. Our findings are manifold. First, we obtain dynamic optimal weights, inversely proportional to volatility. Second, we show that both co-jumps frequency and intensity play a crucial role, as they considerably limit potential losses in the investors' wealth. Third, we prove that jumps in precision reinforce the effect of jumps in price, further reducing optimal allocation. Finally, we highlight how co-jumps may influence investors' choices regarding intertemporal consumption.High-dimensional sparse index tracking based on a multi-step convex optimization approachhttps://zbmath.org/1522.912512023-12-07T16:00:11.105023Z"Shi, Fangquan"https://zbmath.org/authors/?q=ai:shi.fangquan"Shu, Lianjie"https://zbmath.org/authors/?q=ai:shu.lianjie"Luo, Yiling"https://zbmath.org/authors/?q=ai:luo.yiling"Huo, Xiaoming"https://zbmath.org/authors/?q=ai:huo.xiaomingSummary: Both convex and non-convex penalties have been widely proposed to tackle the sparse index tracking problem. Owing to their good property of generating sparse solutions, penalties based on the least absolute shrinkage and selection operator (LASSO) and its variations are often suggested in the stream of convex penalties. However, the LASSO-type penalty is often shown to have poor out-of-sample performance, due to the relatively large biases introduced in the estimates of tracking portfolio weights by shrinking the parameter estimates toward to zero. On the other hand, non-convex penalties could be used to improve the bias issue of LASSO-type penalty. However, the resulting problem is non-convex optimization and thus is computationally intensive, especially in high-dimensional settings. Aimed at ameliorating bias introduced by LASSO-type penalty while preserving computational efficiency, this paper proposes a multi-step convex optimization approach based on the multi-step weighted LASSO (MSW-LASSO) for sparse index tracking. Empirical results show that the proposed method can achieve smaller out-of-sample tracking errors than those based on LASSO-type penalties and have performance competitive to those based on non-convex penalties.Perturbation analysis of sub/super hedging problemshttps://zbmath.org/1522.912592023-12-07T16:00:11.105023Z"Badikov, Sergey"https://zbmath.org/authors/?q=ai:badikov.sergey"Davis, Mark H. A."https://zbmath.org/authors/?q=ai:davis.mark-h-a"Jacquier, Antoine"https://zbmath.org/authors/?q=ai:jacquier.antoineSummary: We investigate the links between various no-arbitrage conditions and the existence of pricing functionals in general markets, and prove the fundamental theorem of asset pricing therein. No-arbitrage conditions, either in this abstract setting or in the case of a market consisting of European call options, give rise to duality properties of infinite-dimensional sub- and super-hedging problems. With a view towards applications, we show how duality is preserved when reducing these problems over finite-dimensional bases. We also introduce a rigorous perturbation analysis of these linear programing problems, and highlight numerically the influence of smile extrapolation on the bounds of exotic options.
{{\copyright} 2021 The Authors. \textit{Mathematical Finance} published by Wiley Periodicals LLC}Techniques for speeding up \(H\)-core protein fittinghttps://zbmath.org/1522.920432023-12-07T16:00:11.105023Z"Ignatov, Andrei"https://zbmath.org/authors/?q=ai:ignatov.andrei"Posypkin, Mikhail"https://zbmath.org/authors/?q=ai:posypkin.mikhail-aSummary: Restoration of the 3D structure of a protein from the sequence of its amino acids (``folding'') is one of the most important and challenging problems in computational biology. The most accurate methods require enormous computational resources due to the large number of variables determining a protein's shape. Coarse-grained models combining several protein atoms into one unified globule partially mitigate this issue. The paper studies one of these models where globules are located in the nodes of the two-dimensional triangular lattice. In this model, folding is reduced to the discrete optimization problem: find positions of protein's globules to maximize the number of contacts between them. We consider a standard procedure that finds an exact solution to this problem. It first generates an \(H\)-core -- a set of positions for hydrophobic globules, which is followed by mapping of protein's hydrophobic globules to these positions by the constraint satisfaction techniques. We propose a way to avoid unnecessary enumeration by skipping infeasible \(H\)-cores prior to mapping. Another contribution of our paper is a procedure that automatically generates constraints to simplify finding the feasible mapping of proteins globules to the lattice nodes. Experiments show that the proposed techniques tremendously accelerate the problem's solving process.
For the entire collection see [Zbl 1508.90001].Long short-term memory network-based wastewater quality prediction model with sparrow search algorithmhttps://zbmath.org/1522.920932023-12-07T16:00:11.105023Z"Li, Guobin"https://zbmath.org/authors/?q=ai:li.guobin"Cui, Qingzhe"https://zbmath.org/authors/?q=ai:cui.qingzhe"Wei, Shengnan"https://zbmath.org/authors/?q=ai:wei.shengnan"Wang, Xiaofeng"https://zbmath.org/authors/?q=ai:wang.xiaofeng.4"Xu, Lixiang"https://zbmath.org/authors/?q=ai:xu.lixiang"He, Lixin"https://zbmath.org/authors/?q=ai:he.lixin"Kwong, Timothy C. H."https://zbmath.org/authors/?q=ai:kwong.timothy-c-h"Tang, Yuanyan"https://zbmath.org/authors/?q=ai:tang.yuanyanSummary: The wastewater treatment process is characterized by uncertainty, non-linearity, time delay and complexity, and is susceptible to many dynamic factors. Since some key water quality parameters are not available in real time, a long short-term memory (LSTM) network water quality prediction model based on sparrow search algorithm (SSA-LSTM) and attention mechanism is proposed to solve the problem. In this model, we take historical data as input, constructs models to learn the feature of internal dynamic changes, introduces the attention mechanism, assigns different weights to the hidden state of the LSTM network by mapping weightings with the learning parameter matrix, and uses the SSA to select the optimal hyperparameters for prediction. As high-latitude feature vectors are subject to the curse of dimension, a PCA-LSTM model is further proposed to apply the principal component analysis (PCA) method to the SSA-LSTM model to reduce the dimensionality of the original data. The SSA-LSTM model without the PCA method (NPCA-LSTM) and the PCA-LSTM model are applied to predict wastewater quality and the PCA-LSTM model shows higher predictive ability.Terminal control of multi-agent systemhttps://zbmath.org/1522.930192023-12-07T16:00:11.105023Z"Antipin, Anatoly"https://zbmath.org/authors/?q=ai:antipin.anatolii-sergeevich"Khoroshilova, Elena"https://zbmath.org/authors/?q=ai:khoroshilova.elena-vladimirovnaSummary: A linear controlled dynamics is considered on a fixed time interval. Dynamics transforms control into a phase trajectory. This trajectory at discrete points of the time interval is loaded with finite-dimensional linear programming problems. These problems define intermediate, initial and boundary value solutions, which correspond to the ends of time subsegments. It is required by the choice of control to form a phase trajectory so that, starting from the initial conditions, the trajectory passes through all solutions of intermediate problems and reaches the terminal conditions at the right end of the time interval. In general, constructions that combine dynamics with mathematical programming problems will be called terminal control problems. The approach to solving these problems is based on the Lagrangian formalism and duality theory. The paper proposes an iterative saddle point computing process for solving the problem of terminal control, which belongs to the class of multi-agent systems. The study was carried out within the framework of evidence-based methodology, i.e. the convergence of computational process with respect to all components of solution is proved.
For the entire collection see [Zbl 1516.90004].Distributed optimization of multi-integrator agent systems with mixed neighbor interactionshttps://zbmath.org/1522.930212023-12-07T16:00:11.105023Z"Chen, Zhao"https://zbmath.org/authors/?q=ai:chen.zhao"Nian, Xiaohong"https://zbmath.org/authors/?q=ai:nian.xiaohong"Meng, Qing"https://zbmath.org/authors/?q=ai:meng.qing|meng.qing.1Summary: In this paper, we consider a constrained distributed optimization problem for multi-integrator agent systems with mixed neighbor interactions, where the neighbors of the agents can be classified as cooperative or noncooperative by introducing a cooperation-competition network into the modeling of the communication graph. The agent's goal is to optimize the total objective function of the cooperators. Since agents can only communicate with their neighbors, they are unable to obtain complete information about cooperators. The distributed optimization algorithm is proposed to classify cooperative subnetworks by introducing additional auxiliary variables, and the outputs of all agents exponentially converge to the optimal solution of the optimization problem under the condition of satisfying the global equality constraint of the subnetwork. The Lyapunov stability theory is used to analyze the algorithm's convergence. Simulation results show the effectiveness of the proposed algorithm in two examples of economic dispatch in smart grids and optimal area coverage in multi-sensor networks.Augmented Lagrangian tracking for distributed optimization with equality and inequality coupling constraintshttps://zbmath.org/1522.930222023-12-07T16:00:11.105023Z"Falsone, Alessandro"https://zbmath.org/authors/?q=ai:falsone.alessandro"Prandini, Maria"https://zbmath.org/authors/?q=ai:prandini.mariaSummary: In this paper we propose a novel augmented Lagrangian tracking distributed optimization algorithm for solving multi-agent optimization problems where each agent has its own decision variables, cost function and constraint set, and the goal is to minimize the sum of the agents' cost functions subject to local constraints plus some additional coupling constraint involving the decision variables of all the agents. In contrast to alternative approaches available in the literature, the proposed algorithm jointly features a constant penalty parameter, the ability to cope with unbounded local constraint sets, and the ability to handle both affine equality and nonlinear inequality coupling constraints, while requiring convexity only. The effectiveness of the approach is shown first on an artificial example with complexity features that make other state-of-the-art algorithms not applicable and then on a realistic example involving the optimization of the charging schedule of a fleet of electric vehicles.Dealing with infeasibility in multi-parametric programming for application to explicit model predictive controlhttps://zbmath.org/1522.930592023-12-07T16:00:11.105023Z"Falsone, Alessandro"https://zbmath.org/authors/?q=ai:falsone.alessandro"Bianchi, Federico"https://zbmath.org/authors/?q=ai:bianchi.federico"Prandini, Maria"https://zbmath.org/authors/?q=ai:prandini.mariaSummary: Motivated by explicit model predictive control, we address infeasibility in multi-parametric quadratic programming according to the exact penalty function approach, where some user-chosen parameter-dependent constraints are relaxed and the 1-norm of their violation is penalized in the cost function. We characterize the relation between the resulting multi-parametric quadratic program and the original one and show that, as the penalty coefficient grows to infinity, the solution to the former provides a piecewise affine continuous function, which is an optimal solution for the latter over the feasibility region, while it minimizes the 1-norm of the relaxed constraints violation over the infeasibility region.Distributed conditional cooperation model predictive control of interconnected microgridshttps://zbmath.org/1522.930622023-12-07T16:00:11.105023Z"Sampathirao, Ajay Kumar"https://zbmath.org/authors/?q=ai:sampathirao.ajay-kumar"Hofmann, Steffen"https://zbmath.org/authors/?q=ai:hofmann.steffen"Raisch, Jörg"https://zbmath.org/authors/?q=ai:raisch.jorg"Hans, Christian Andreas"https://zbmath.org/authors/?q=ai:hans.christian-andreasSummary: In this paper, we propose a model predictive control based operation strategy that allows for power exchange between interconnected microgrids. Particularly, the approach ensures that each microgrid benefits from power exchange with others. This is realised by including a condition which is based on the islanded operation cost. The overall model predictive control problem is posed as a mixed-integer quadratically-constrained program and solved using a distributed algorithm that iteratively updates continuous and integer variables. For this algorithm, termination, feasibility and computational properties are discussed. The performance and the computational benefits of the proposed strategy are highlighted in an illustrative case study.Robust observer design and Nash-game-driven fuzzy optimization for uncertain dynamical systemshttps://zbmath.org/1522.930692023-12-07T16:00:11.105023Z"Hu, Zhanyi"https://zbmath.org/authors/?q=ai:hu.zhanyi"Huang, Jin"https://zbmath.org/authors/?q=ai:huang.jin"Yang, Zeyu"https://zbmath.org/authors/?q=ai:yang.zeyu"Zhong, Zhihua"https://zbmath.org/authors/?q=ai:zhong.zhihuaSummary: The estimation of system state is pivotal especially when system state can't be directly measured. In this paper, an optimal robust observer is proposed for uncertain dynamical systems. The fuzzy set theory is utilized to interpret the uncertainty bound. Unlike fuzzy-logic based observer, the proposed observer is not IF-THEN fuzzy rules-based. The uniform boundedness and ultimate uniform boundedness of estimation error are guaranteed. Then, the optimal design problem of the two tunable parameters is formulated under the framework of a two-player Nash game. The fuzzy-set theoretic cost functions of the two players are associated with the observer performance. We prove that the Nash equilibria does exist, and provide procedures in solving this optimal problem. Numerical simulations are conducted on an inverted pendulum using MATLAB/Simulink. The theory is illustrated by the observer design for an inverted pendulum subject to the earthquake excitation. The novelty of this work is an integrated framework for optimal robust observer design, which creatively blends the state estimation theory, the fuzzy set theory and the Nash game theory.Adaptive optimal control of continuous-time nonlinear affine systems via hybrid iterationhttps://zbmath.org/1522.930952023-12-07T16:00:11.105023Z"Qasem, Omar"https://zbmath.org/authors/?q=ai:qasem.omar"Gao, Weinan"https://zbmath.org/authors/?q=ai:gao.weinan"Vamvoudakis, Kyriakos G."https://zbmath.org/authors/?q=ai:vamvoudakis.kyriakos-gSummary: In this paper, a novel successive approximation framework, named hybrid iteration (HI), is proposed to fill up the performance gap between two well-known dynamic programming algorithms, namely policy iteration (PI) and value iteration (VI). Using HI, an approximated optimal control policy can be learned without prior knowledge of an initial admissible control policy required by PI. Additionally, the HI algorithm converges to the optimal solution much faster than VI, and thus requires tremendously less number of learning iterations and CPU-time, compared to VI. Initially, we develop a model-based HI algorithm, and then extend it to a data-driven HI algorithm which learns the optimal control policy without any information of the physics of the system. Simulation results demonstrate the efficacy of the proposed HI algorithm.Supervisory adaptive interval type-2 fuzzy sliding mode control for planar cable-driven parallel robots using Grasshopper optimizationhttps://zbmath.org/1522.930982023-12-07T16:00:11.105023Z"Aghaseyedabdollah, Mh."https://zbmath.org/authors/?q=ai:aghaseyedabdollah.mh"Abedi, M."https://zbmath.org/authors/?q=ai:abedi.mohammad|abedi.mostafa|abedi.maryam|abedi.majid|abedi.mohsen"Pourgholi, M."https://zbmath.org/authors/?q=ai:pourgholi.mahdi(no abstract)System identification of fuzzy relation matrix models by semi-tensor product operationshttps://zbmath.org/1522.931072023-12-07T16:00:11.105023Z"Lyu, Hong L."https://zbmath.org/authors/?q=ai:lyu.hongli"Wang, Wilson"https://zbmath.org/authors/?q=ai:wang.wilson"Liu, Xiao P."https://zbmath.org/authors/?q=ai:liu.xiaopingSummary: In order to facilitate the representation of fuzzy relation matrix (FRM) models, a new system identification technique is proposed in this work to recognize the architecture and parameters of FRM models based on the semi-tensor product (STP) operation. Firstly, a fuzzy STP algorithm is defined for fuzzy inference. Secondly, a novel FRM framework is proposed for system parameter identification. Thirdly, the recognized FRM parameters are optimized to improve fuzzy system performance by the use of a hybrid training method based on the least squares estimator and the recursive Levenberg-Marquaedt algorithm. The effectiveness of the proposed structure and parameter identification techniques is verified by simulation of a multi-steps-ahead prediction modeling. Simulation results show that the proposed fuzzy STP technology is efficient for system identification, and the proposed matrix expression can be used to design multi-input multi-output (MIMO) systems with fuzzy FRM models.Embedding active learning in batch-to-batch optimization using reinforcement learninghttps://zbmath.org/1522.931332023-12-07T16:00:11.105023Z"Byun, Ha-Eun"https://zbmath.org/authors/?q=ai:byun.ha-eun"Kim, Boeun"https://zbmath.org/authors/?q=ai:kim.boeun"Lee, Jay H."https://zbmath.org/authors/?q=ai:lee.jay-hSummary: Batch-to-batch (B2B) or run-to-run (R2R) optimization refers to the strategy of updating the operating parameters of a batch run based on the results of previous runs and exploits the repetitive nature of batch process operation. Although B2B optimization uses feedback from previous batch runs to learn about model uncertainty and improve the operation of future runs, the standard techniques have the limitations of passive learning and being myopic in making adjustments. This work proposes a novel way to use the reinforcement learning approach to embed the active learning feature into B2B optimization. For this, the B2B optimization problem is formulated as a maximization of a long-term performance of repeated batch runs, which are modeled as a stochastic process with uncertain parameters. To solve the resulting Bayes-adaptive Markov decision process (BAMDP) problem in a near-optimal manner, a policy gradient reinforcement learning algorithm is employed. Through case studies, the behavior and effectiveness of the proposed B2B optimization method are examined by comparing it with the traditional certainty equivalence based B2B optimization method with passive learning.Likelihood landscape and maximum likelihood estimation for the discrete orbit recovery modelhttps://zbmath.org/1522.940052023-12-07T16:00:11.105023Z"Fan, Zhou"https://zbmath.org/authors/?q=ai:fan.zhou"Sun, Yi"https://zbmath.org/authors/?q=ai:sun.yi"Wang, Tianhao"https://zbmath.org/authors/?q=ai:wang.tianhao"Wu, Yihong"https://zbmath.org/authors/?q=ai:wu.yihongSummary: We study the nonconvex optimization landscape for maximum likelihood estimation in the discrete orbit recovery model with Gaussian noise. This is a statistical model motivated by applications in molecular microscopy and image processing, where each measurement of an unknown object is subject to an independent random rotation from a known rotational group. Equivalently, it is a Gaussian mixture model where the mixture centers belong to a group orbit.
We show that fundamental properties of the likelihood landscape depend on the signal-to-noise ratio and the group structure. At low noise, this landscape is ``benign'' for any discrete group, possessing no spurious local optima and only strict saddle points. At high noise, this landscape may develop spurious local optima, depending on the specific group. We discuss several positive and negative examples, and provide a general condition that ensures a globally benign landscape at high noise. For cyclic permutations of coordinates on \(\mathbb{R}^d\) (multireference alignment), there may be spurious local optima when \(d \geq 6\), and we establish a correspondence between these local optima and those of a surrogate function of the phase variables in the Fourier domain.
We show that the Fisher information matrix transitions from resembling that of a single Gaussian distribution in low noise to having a graded eigenvalue structure in high noise, which is determined by the graded algebra of invariant polynomials under the group action. In a local neighborhood of the true object, where the neighborhood size is independent of the signal-to-noise ratio, the landscape is strongly convex in a reparametrized system of variables given by a transcendence basis of this polynomial algebra. We discuss implications for optimization algorithms, including slow convergence of expectation-maximization, and possible advantages of momentum-based acceleration and variable reparametrization for first- and second-order descent methods.
{{\copyright} 2021 Wiley Periodicals LLC.}Sparse signal reconstruction via collaborative neurodynamic optimizationhttps://zbmath.org/1522.940122023-12-07T16:00:11.105023Z"Che, Hangjun"https://zbmath.org/authors/?q=ai:che.hangjun"Wang, Jun"https://zbmath.org/authors/?q=ai:wang.jun.1"Cichocki, Andrzej"https://zbmath.org/authors/?q=ai:cichocki.andrzejSummary: In this paper, we formulate a mixed-integer problem for sparse signal reconstruction and reformulate it as a global optimization problem with a surrogate objective function subject to underdetermined linear equations. We propose a sparse signal reconstruction method based on collaborative neurodynamic optimization with multiple recurrent neural networks for scattered searches and a particle swarm optimization rule for repeated repositioning. We elaborate on experimental results to demonstrate the outperformance of the proposed approach against ten state-of-the-art algorithms for sparse signal reconstruction.Sparse estimation: an MMSE approachhttps://zbmath.org/1522.940142023-12-07T16:00:11.105023Z"Pang, Tongyao"https://zbmath.org/authors/?q=ai:pang.tongyao"Shen, Zuowei"https://zbmath.org/authors/?q=ai:shen.zuoweiSummary: The objective of this paper is to estimate parameters with a sparse prior via the minimum mean square error (MMSE) approach. We model the sparsity by the Bernoulli-uniform prior. The MMSE estimator gives the posterior mean of the parameter to be estimated. However, its computation involves multiple integrations of many variables that is hard to implement numerically. In order to overcome this difficulty, we develop a coordinate minimization algorithm to approximate the MMSE estimator for any arbitrary given prior. We connect this algorithm to a variational model and establish a comprehensive convergence analysis. The algorithm converges to a special stationary point of the variational model, which attains the minimum of the mean square error at each coordinate when others are fixed. Then, this general algorithm is applied to the Bernoulli-uniform sparse prior and leads to a stable estimator that provides a good balance between sparsity and unbiasedness. The advantages of our sparsity model and algorithm over other approaches (e.g., the maximum a posteriori approaches) are analysed in detail and further demonstrated by numerical simulations. The applications of the general theory and algorithm developed here go beyond sparse estimation.Framework for segmented threshold \(\ell_0\) gradient approximation based network for sparse signal recoveryhttps://zbmath.org/1522.940162023-12-07T16:00:11.105023Z"Vivekanand, V."https://zbmath.org/authors/?q=ai:vivekanand.v"Mishra, Deepak"https://zbmath.org/authors/?q=ai:mishra.deepakSummary: Signal reconstruction from compressed sensed data need iterative methods since the sparse measurement matrix is analytically non invertible. The iterative thresholding and \(\ell_0\) function minimization are of special interest as these two operations provide sparse solution. However these methods need an inverse operation corresponding to the measurement matrix for estimating the reconstruction error. The pseudo-inverse of the measurement matrix is used in general for this purpose. Here a sparse signal recovery framework using an approximate inverse matrix \(\mathcal{Q}\) and iterative segment thresholding of \(\ell_0\) and \(\ell_1\) norm with residue addition is presented. Two recovery algorithms are developed using this framework. The \(\ell_0\) based method is later developed to a basis function dictionary based network for sparse signal recovery. The proposed framework enables the users experiment with different inverse matrix to achieve better efficiency in sparse signal recovery and implement the algorithm in computationally efficient way.Fully automated differential-linear attacks against ARX ciphershttps://zbmath.org/1522.940382023-12-07T16:00:11.105023Z"Bellini, Emanuele"https://zbmath.org/authors/?q=ai:bellini.emanuele"Gerault, David"https://zbmath.org/authors/?q=ai:gerault.david"Grados, Juan"https://zbmath.org/authors/?q=ai:grados.juan"Makarim, Rusydi H."https://zbmath.org/authors/?q=ai:makarim.rusydi-h"Peyrin, Thomas"https://zbmath.org/authors/?q=ai:peyrin.thomasSummary: In this paper, we present a fully automated tool for differential-linear attacks using Mixed-Integer Linear Programming (MILP) and Mixed-Integer Quadratic Constraint Programming (MIQCP) techniques, which is, to the best of our knowledge, the very first attempt to fully automate such attacks. We use this tool to improve the correlations of the best 9 and 10-round differential-linear distinguishers on \texttt{speck32/64}, and reach 11 rounds for the first time. Furthermore, we improve the latest 14-round key-recovery attack against \texttt{speck32/64}, using differential-linear distinguishers obtained with our MILP/MIQCP tool. The techniques we present are generic and can be applied to other ARX ciphers as well.
For the entire collection see [Zbl 1521.94005].