Recent zbMATH articles in MSC 49https://zbmath.org/atom/cc/492022-11-17T18:59:28.764376ZUnknown authorWerkzeugExistence-uniqueness of positive solutions to nonlinear impulsive fractional differential systems and optimal controlhttps://zbmath.org/1496.340502022-11-17T18:59:28.764376Z"Song, Shu"https://zbmath.org/authors/?q=ai:song.shu"Zhang, Lingling"https://zbmath.org/authors/?q=ai:zhang.lingling"Zhou, Bibo"https://zbmath.org/authors/?q=ai:zhou.bibo"Zhang, Nan"https://zbmath.org/authors/?q=ai:zhang.nan.1Summary: In this thesis, we investigate a kind of impulsive fractional order differential systems involving control terms. By using a class of \(\varphi \)-concave-convex mixed monotone operator fixed point theorem, we obtain a theorem on the existence and uniqueness of positive solutions for the impulsive fractional differential equation, and the optimal control problem of positive solutions is also studied. As applications, an example is offered to illustrate our main results.Turnpike properties of solutions of a differential inclusion with a Lyapunov function. IIhttps://zbmath.org/1496.340932022-11-17T18:59:28.764376Z"Zaslavski, Alexander J."https://zbmath.org/authors/?q=ai:zaslavski.alexander-jSummary: We study the turnpike phenomenon for approximate solutions of optimal problems governed by a differential inclusion with a Lyapunov function. This differential inclusion generates a dynamical system which has a prototype in mathematical economics. In our previous research we obtained turnpike results for a collection of approximate optimal trajectories with a fixed initial point. In the present paper, under a certain assumption, we extend these results for all approximate optimal trajectories.
For Part I see [ibid. 7, No. 3, 1085--1102 (2022; Zbl 1490.34061)].Homogenization of boundary optimal control problemhttps://zbmath.org/1496.350462022-11-17T18:59:28.764376Z"Mishra, Indira"https://zbmath.org/authors/?q=ai:mishra.indiraSummary: In this article, we study the asymptotic behavior of solutions to some optimal control problems, governed by an elliptic boundary value problem with Robin boundary conditions in a periodically perforated domain. The coefficients of the differential operator in the state equation and in the cost-functional are rapidly oscillating. We also study the boundary homogenization of some optimal control problems.Gradient flows and nonlinear power methods for the computation of nonlinear eigenfunctionshttps://zbmath.org/1496.350672022-11-17T18:59:28.764376Z"Bungert, Leon"https://zbmath.org/authors/?q=ai:bungert.leon"Burger, Martin"https://zbmath.org/authors/?q=ai:burger.martinSummary: This chapter describes how gradient flows and nonlinear power methods in Banach spaces can be used to solve nonlinear eigenvector-dependent eigenvalue problems, and how these can be approximated using \(\Gamma\)-convergence. We review several flows from literature, which were proposed to compute nonlinear eigenfunctions, and show that they all relate to normalized gradient flows. Furthermore, we show that the implicit Euler discretization of gradient flows gives rise to a nonlinear power method of the proximal operator and we demonstrate their convergence to nonlinear eigenfunctions. Finally, we prove that \(\Gamma\)-convergence of functionals implies convergence of their ground states.
For the entire collection see [Zbl 1492.49003].4-harmonic functions and beyondhttps://zbmath.org/1496.350792022-11-17T18:59:28.764376Z"Grecu, A."https://zbmath.org/authors/?q=ai:grecu.alexandru-tudor|grecu.andreea|grecu.andrei"Mihailescu, M."https://zbmath.org/authors/?q=ai:mihailescu.marius-iulian|mihailescu.mihail|mihailescu.mihaiSummary: The family of partial differential equations \(-\Delta_4 u - \varepsilon \Delta_\infty u = 0 (\varepsilon > 0)\) is studied in a bounded domain \(\Omega\) for given boundary data. We show that for each \(\varepsilon > 0\) the problem has a unique viscosity solution which is exactly the \((4+ \varepsilon)\)-harmonic map with the given boundary data. We also explore the connections between the solutions of these problems and infinity harmonic and 4-harmonic maps by studying the limiting behavior of the solutions as \(\varepsilon \rightarrow \infty\) and \(\varepsilon \rightarrow 0^+\), respectively.Boundary regularity of minimal oriented hypersurfaces on a manifoldhttps://zbmath.org/1496.351352022-11-17T18:59:28.764376Z"Steinbrüchel, Simone"https://zbmath.org/authors/?q=ai:steinbruchel.simoneSummary: In this article we prove that all boundary points of a minimal oriented hypersurface in a Riemannian manifold are regular, that is, in a neighborhood of any boundary point, the minimal surface is a \(C^{1, \frac{1}{4}}\) submanifold with boundary.An optimal insulation problemhttps://zbmath.org/1496.351892022-11-17T18:59:28.764376Z"Pietra, Francesco Della"https://zbmath.org/authors/?q=ai:della-pietra.francesco"Nitsch, Carlo"https://zbmath.org/authors/?q=ai:nitsch.carlo"Trombetti, Cristina"https://zbmath.org/authors/?q=ai:trombetti.cristinaAuthors' abstract: In this paper we consider a minimization problem which arises from thermal insulation. A compact connected set \(K,\) which represents a conductor of constant temperature, say \(1,\) is thermally insulated by surrounding it with a layer of thermal insulator, the open set \(\Omega\setminus K\) with \(K\subset \overline{\Omega}.\) The heat dispersion is then obtained as \[ \inf \left\{ \int_{\Omega}|\nabla\varphi|^2dx +\beta \int_{\partial^\ast\Omega}\varphi^2d\mathcal{H}^{n-1},\quad\varphi \in H^1(\mathbb{R}^n),\quad \varphi \geq 1\ \text{in}\ K\right\}, \] for some positive constant \(\beta.\)
We mostly restrict our analysis to the case of an insulating layer of constant thickness. We let the set \(K\) vary, under prescribed geometrical constraints, and we look for the best (or worst) geometry in terms of heat dispersion. We show that under perimeter constraint the disk in two dimensions is the worst one. The same is true for the ball in higher dimension but under different constraints. We finally discuss few open problems.
Reviewer: Dian K. Palagachev (Bari)Constrained problems via sub-supersolutionhttps://zbmath.org/1496.352202022-11-17T18:59:28.764376Z"Motreanu, Dumitru"https://zbmath.org/authors/?q=ai:motreanu.dumitruThis article discusses a class of elliptic quasilinear differential equations involving the \((p,q)\)-Laplace operator and a convection-convolution term. The author devices a sub-super solution method in order to show the existence of a weak solution in an appropriate Sobolev space.
Reviewer: Marius Ghergu (Dublin)Local null controllability of a class of non-Newtonian incompressible viscous fluidshttps://zbmath.org/1496.353132022-11-17T18:59:28.764376Z"de Carvalho, Pitágoras Pinheiro"https://zbmath.org/authors/?q=ai:de-carvalho.pitagoras-pinheiro"Límaco, Juan"https://zbmath.org/authors/?q=ai:limaco.juan"Menezes, Denilson"https://zbmath.org/authors/?q=ai:menezes.denilson"Thamsten, Yuri"https://zbmath.org/authors/?q=ai:thamsten.yuriSummary: We investigate the null controllability property of systems that mathematically describe the dynamics of some non-Newtonian incompressible viscous flows. The principal model we study was proposed by O. A. Ladyzhenskaya, although the techniques we develop here apply to other fluids having a shear-dependent viscosity. Taking advantage of the Pontryagin Minimum Principle, we utilize a bootstrapping argument to prove that sufficiently smooth controls to the forced linearized Stokes problem exist, as long as the initial data in turn has enough regularity. From there, we extend the result to the nonlinear problem. As a byproduct, we devise a quasi-Newton algorithm to compute the states and a control, which we prove to converge in an appropriate sense. We finish the work with some numerical experiments.Variational and stability properties of coupled NLS equations on the star graphhttps://zbmath.org/1496.353552022-11-17T18:59:28.764376Z"Cely, Liliana"https://zbmath.org/authors/?q=ai:cely.liliana"Goloshchapova, Nataliia"https://zbmath.org/authors/?q=ai:goloshchapova.nataliiaSummary: We consider variational and stability properties of a system of two coupled nonlinear Schrödinger equations on the star graph \(\Gamma\) with the \(\delta\) coupling at the vertex of \(\Gamma\). The first part is devoted to the proof of an existence of the ground state as the minimizer of the constrained energy in the cubic case. This result extends the one obtained recently for the coupled NLS equations on the line.
In the second part, we study stability properties of several families of standing waves in the case of a general power nonlinearity. In particular, we consider one-component standing waves \(e^{i \omega t} (\Phi_1 (x), 0)\) and \(e^{i \omega t} (0, \Phi_2 (x))\). Moreover, we study two-component standing waves \(e^{i \omega t} (\Phi (x), \Phi (x))\) for the case of power nonlinearity depending on a unique power parameter \(p\).
To our knowledge, these are the first results on variational and stability properties of coupled NLS equations on graphs.On Chien's question to the Hu-Washizu three-field functional and variational principlehttps://zbmath.org/1496.353812022-11-17T18:59:28.764376Z"Sun, Bohua"https://zbmath.org/authors/?q=ai:sun.bohua(no abstract)Shape optimization of a thermal insulation problemhttps://zbmath.org/1496.353832022-11-17T18:59:28.764376Z"Bucur, Dorin"https://zbmath.org/authors/?q=ai:bucur.dorin"Nahon, Mickaël"https://zbmath.org/authors/?q=ai:nahon.mickael"Nitsch, Carlo"https://zbmath.org/authors/?q=ai:nitsch.carlo"Trombetti, Cristina"https://zbmath.org/authors/?q=ai:trombetti.cristinaSummary: We study a shape optimization problem involving a solid \(K\subset\mathbb{R}^n\) that is maintained at constant temperature and is enveloped by a layer of insulating material \(\Omega\) which obeys a generalized boundary heat transfer law. We minimize the energy of such configurations among all \((K, \Omega)\) with prescribed measure for \(K\) and \(\Omega\), and no topological or geometrical constraints. In the convection case (corresponding to Robin boundary conditions on \(\partial\Omega\)) we obtain a full description of minimizers, while for general heat transfer conditions, we prove the existence and regularity of solutions and give a partial description of minimizers.Correction to: ``Second order local minimal-time mean field games''https://zbmath.org/1496.353912022-11-17T18:59:28.764376Z"Ducasse, Romain"https://zbmath.org/authors/?q=ai:ducasse.romain"Mazanti, Guilherme"https://zbmath.org/authors/?q=ai:mazanti.guilherme"Santambrogio, Filippo"https://zbmath.org/authors/?q=ai:santambrogio.filippoCorrection to the authors' paper [ibid. 29, No. 4, Paper No. 37, 32 p. (2022; Zbl 1492.35352)].A Fokker-Planck feedback control framework for optimal personalized therapies in colon cancer-induced angiogenesishttps://zbmath.org/1496.353992022-11-17T18:59:28.764376Z"Roy, Souvik"https://zbmath.org/authors/?q=ai:roy.souvik.1|roy.souvik"Pan, Zui"https://zbmath.org/authors/?q=ai:pan.zui"Pal, Suvra"https://zbmath.org/authors/?q=ai:pal.suvraSummary: In this paper, a new framework for obtaining personalized optimal treatment strategies in colon cancer-induced angiogenesis is presented. The dynamics of colon cancer is given by a Itó stochastic process, which helps in modeling the randomness present in the system. The stochastic dynamics is then represented by the Fokker-Planck (FP) partial differential equation that governs the evolution of the associated probability density function. The optimal therapies are obtained using a three step procedure. First, a finite dimensional FP-constrained optimization problem is formulated that takes input individual noisy patient data, and is solved to obtain the unknown parameters corresponding to the individual tumor characteristics. Next, a sensitivity analysis of the optimal parameter set is used to determine the parameters to be controlled, thus, helping in assessing the types of treatment therapies. Finally, a feedback FP control problem is solved to determine the optimal combination therapies. Numerical results with the combination drug, comprising of Bevacizumab and Capecitabine, demonstrate the efficiency of the proposed framework.Gradient flow formulation of diffusion equations in the Wasserstein space over a metric graphhttps://zbmath.org/1496.354052022-11-17T18:59:28.764376Z"Erbar, Matthias"https://zbmath.org/authors/?q=ai:erbar.matthias"Forkert, Dominik"https://zbmath.org/authors/?q=ai:forkert.dominik"Maas, Jan"https://zbmath.org/authors/?q=ai:maas.jan"Mugnolo, Delio"https://zbmath.org/authors/?q=ai:mugnolo.delioSummary: This paper contains two contributions in the study of optimal transport on metric graphs. Firstly, we prove a Benamou-Brenier formula for the Wasserstein distance, which establishes the equivalence of static and dynamical optimal transport. Secondly, in the spirit of Jordan-Kinderlehrer-Otto, we show that McKean-Vlasov equations can be formulated as gradient flow of the free energy in the Wasserstein space of probability measures. The proofs of these results are based on careful regularisation arguments to circumvent some of the difficulties arising in metric graphs, namely, branching of geodesics and the failure of semi-convexity of entropy functionals in the Wasserstein space.Optimal shapes for tree rootshttps://zbmath.org/1496.354072022-11-17T18:59:28.764376Z"Bressan, Alberto"https://zbmath.org/authors/?q=ai:bressan.alberto"Galtung, Sondre T."https://zbmath.org/authors/?q=ai:galtung.sondre-tesdal"Sun, Qing"https://zbmath.org/authors/?q=ai:sun.qingFractional truncated Laplacians: representation formula, fundamental solutions and applicationshttps://zbmath.org/1496.354202022-11-17T18:59:28.764376Z"Birindelli, Isabeau"https://zbmath.org/authors/?q=ai:birindelli.isabeau"Galise, Giulio"https://zbmath.org/authors/?q=ai:galise.giulio"Topp, Erwin"https://zbmath.org/authors/?q=ai:topp.erwinSummary: We introduce some nonlinear extremal nonlocal operators that approximate the, so called, truncated Laplacians. For these operators we construct representation formulas that lead to the construction of what, with an abuse of notation, could be called ``fundamental solutions''. This, in turn, leads to Liouville type results. The interest is double: on one hand we wish to ``understand'' what is the right way to define the nonlocal version of the truncated Laplacians, on the other, we introduce nonlocal operators whose nonlocality is on one dimensional lines, and this dramatically changes the prospective, as is quite clear from the results obtained that often differ significantly with the local case or with the case where the nonlocality is diffused. Surprisingly this is true also for operators that approximate the Laplacian.Divergence \& curl with fractional orderhttps://zbmath.org/1496.354342022-11-17T18:59:28.764376Z"Liu, Liguang"https://zbmath.org/authors/?q=ai:liu.liguang"Xiao, Jie"https://zbmath.org/authors/?q=ai:xiao.jie.1Summary: This paper presents a novel analysis for Function Space Norms (F.S.N.) \& Partial Differential Equations (P.D.E.) within the fractional-nonlocal pair \(\{\operatorname{div}^*\mathbf{v},\operatorname{curl}^*\mathbf{v}\}\) that extends the classical-local pair \(\{\operatorname{div}\mathbf{v},\operatorname{curl}\mathbf{v}\}\) which has an inherent physical
content because of causing the conservation of mass \& the rotation produced by fluid elements in motion.Optimal stochastic forcings for sensitivity analysis of linear dynamical systemshttps://zbmath.org/1496.370062022-11-17T18:59:28.764376Z"Nechepurenko, Yuri M."https://zbmath.org/authors/?q=ai:nechepurenko.yuri-m"Zasko, Grigory V."https://zbmath.org/authors/?q=ai:zasko.grigory-vSummary: The paper is devoted to the construction of optimal stochastic forcings for studying the sensitivity of linear dynamical systems to external perturbations. The optimal forcings are sought to maximize the Schatten norms of the response. As an example, we consider the problem of constructing the optimal stochastic forcing for the linear dynamical system arising from the analysis of large-scale structures in a stratified turbulent Couette flow.Turnpike properties for discrete-time optimal control problems with a Lyapunov functionhttps://zbmath.org/1496.370112022-11-17T18:59:28.764376Z"Zaslavski, Alexander J."https://zbmath.org/authors/?q=ai:zaslavski.alexander-jSummary: We study the turnpike phenomenon for discrete disperse dynamical systems introduced in [Sib. Mat. Zh. 21, No. 4, 136--145 (1980; Zbl 0453.90024)] by \textit{A. M. Rubinov}, which have a prototype in mathematical economics.Rigidity of the Pu inequality and quadratic isoperimetric constants of normed spaceshttps://zbmath.org/1496.460112022-11-17T18:59:28.764376Z"Creutz, Paul"https://zbmath.org/authors/?q=ai:creutz.paulThe author furnishes an enhanced bound on the filling areas curves (not closed geodesics) in Banach spaces. He shows rigidity of \textit{P. M. Pu}'s classical systolic inequality [Pac. J. Math. 2, 55--71 (1952; Zbl 0046.39902)] and examines the isoperimetric constants of normed spaces.
Reviewer: Mohammed El Aïdi (Bogotá)Improving semigroup bounds with resolvent estimateshttps://zbmath.org/1496.470672022-11-17T18:59:28.764376Z"Helffer, B."https://zbmath.org/authors/?q=ai:helffer.bernard"Sjöstrand, J."https://zbmath.org/authors/?q=ai:sjostrand.johannesThe authors revisit the proof of the Gearhart-Prüss-Huang-Greiner-theorem for a semigroup \((S(t))_{t\geq 0}\), following the general idea of the proofs that have been given in the literature and to get an explicit estimate on \(\|S(t)\|\) in terms of bounds on the resolvent of the generator. A~first version of this paper was presented by the two authors in [``From resolvent bounds to semigroup bounds'', Preprint (2010), \url{arXiv:1001.4171}] together with applications in semi-classical analysis and some of these results has been subsequently published in two books written by the authors. Their aim in the paper under review is to present new improvements, partially motivated by a paper of \textit{D.-Y. Wei} [Sci. China, Math. 64, No. 3, 507--518 (2021; Zbl 1464.35260)]. Along the way, they discuss optimization problems confirming the optimality of their results.
Reviewer: Sven-Ake Wegner (Hamburg)On the construction of maximal \(p\)-cyclically monotone operatorshttps://zbmath.org/1496.470772022-11-17T18:59:28.764376Z"Bueno, Orestes"https://zbmath.org/authors/?q=ai:bueno.orestes"Cotrina, John"https://zbmath.org/authors/?q=ai:cotrina.johnThe authors describe a method for the construction of explicit examples of maximal \(p\)-cyclically monotone operators. Some new examples of such operators with \(p=2\) and \(p=3\) are also presented.
Reviewer: Rodica Luca (Iaşi)On solution sets of nonlinear equations with nonsmooth operators in Hilbert space and the quasi-solution methodhttps://zbmath.org/1496.470882022-11-17T18:59:28.764376Z"Kokurin, Mikhail Yu."https://zbmath.org/authors/?q=ai:kokurin.mihail-yu|kokurin.mikhail-yurjevichThe author investigates nonlinear irregular equations in Hilbert space with a~priori constraints, without assuming differentiability for the governing operator. The constraints are described by a bounded closed set that is part of an extended source representation class expressed in terms of a given linear operator. The unique solvability of the problem is not assumed. It is established that solutions to the problem form a cluster of diameter strictly less than the one of the constraints set. The approximation properties of the quasi-solution method are addressed in relation to the solution set of the original problem.
Reviewer: Radu Ioan Boţ (Wien)Some remarks on regularized nonconvex variational inequalitieshttps://zbmath.org/1496.470892022-11-17T18:59:28.764376Z"Balooee, Javad"https://zbmath.org/authors/?q=ai:balooee.javad"Kim, Jong Kyu"https://zbmath.org/authors/?q=ai:kim.jong-kyuSummary: In this paper, we investigate and analyze the nonconvex variational inequalities introduced by \textit{M. A. Noor} in [Optim. Lett. 3, No. 3, 411--418 (2009; Zbl 1171.58307)] and [Comput. Math. Model. 21, No. 1, 97--108 (2010; Zbl 1201.65114)] and prove that the algorithms and results in the above mentioned papers are not valid. To overcome the problems in the above cited papers, we introduce and consider a new class of variational inequalities, named regularized nonconvex variational inequalities, instead of the class of nonconvex variational inequalities introduced in the above mentioned papers. We also consider a class of nonconvex Wiener-Hopf equations and establish the equivalence between the regularized nonconvex variational inequalities and the fixed point problems as well as the nonconvex Wiener-Hopf equations. By using the obtained equivalence formulations, we prove the existence of a unique solution for the regularized nonconvex variational inequalities and propose some projection iterative schemes for solving the regularized nonconvex variational inequalities. We also study the convergence analysis of the suggested iterative schemes under some certain conditions.Set-valued mixed quasi-equilibrium problems with operator solutionshttps://zbmath.org/1496.470902022-11-17T18:59:28.764376Z"Ram, Tirth"https://zbmath.org/authors/?q=ai:ram.tirth"Khanna, Anu Kumari"https://zbmath.org/authors/?q=ai:khanna.anu-kumari"Kour, Ravdeep"https://zbmath.org/authors/?q=ai:kour.ravdeepSummary: In this paper, we introduce and study generalized mixed operator quasi-equilibrium problems (GMQOEP) in Hausdorff topological vector spaces and prove the existence results for the solution of (GMQOEP) in compact and noncompact settings by employing 1-person game theorems. Moreover, using coercive condition, hemicontinuity of the functions and KKM theorem, we prove new results on the existence of solution for the particular case of (GMQOEP), that is, generalized mixed operator equilibrium problem (GMOEP).Parametric generalized multi-valued nonlinear quasi-variational inclusion problemhttps://zbmath.org/1496.470912022-11-17T18:59:28.764376Z"Khan, F. A."https://zbmath.org/authors/?q=ai:khan.faizan-ahmad"Alanazi, A. M."https://zbmath.org/authors/?q=ai:alanazi.abdulaziz-m"Ali, Javid"https://zbmath.org/authors/?q=ai:ali.javid"Alanazi, Dalal J."https://zbmath.org/authors/?q=ai:alanazi.dalal-jSummary: In this paper, we investigate the behavior and sensitivity analysis of a solution set for a parametric generalized multi-valued nonlinear quasi-variational inclusion problem in a real Hilbert space. For this study, we utilize the technique of resolvent operator and the property of a fixed-point set of a multi-valued contractive mapping. We also examine Lipschitz continuity of the solution set with respect to the parameter under some appropriate conditions.General convergence analysis of projection methods for a system of variational inequalities in \(q\)-uniformly smooth Banach spaceshttps://zbmath.org/1496.470942022-11-17T18:59:28.764376Z"Gong, Qian-Fen"https://zbmath.org/authors/?q=ai:gong.qianfen"Wen, Dao-Jun"https://zbmath.org/authors/?q=ai:wen.daojunSummary: In this paper, we introduce and consider a system of variational inequalities involving two different operators in \(q\)-uniformly smooth Banach spaces. We suggest and analyze a new explicit projection method for solving the system under some more general conditions. Our results extend and unify the results of \textit{R. U. Verma} [Appl. Math. Lett. 18, No. 11, 1286--1292 (2005; Zbl 1099.47054)] and Yao, Liou and Kang [\textit{Y.-H. Yao} et al., J. Glob. Optim. 55, No. 4, 801--811 (2013; Zbl 1260.47085)] and some other previously known results.Viscosity approximation methods for hierarchical optimization problems in \(\mathrm{CAT}(0)\) spaceshttps://zbmath.org/1496.470982022-11-17T18:59:28.764376Z"Liu, Xin-Dong"https://zbmath.org/authors/?q=ai:liu.xindong"Chang, Shih-Sen"https://zbmath.org/authors/?q=ai:chang.shih-senSummary: This paper aims at investigating viscosity approximation methods for solving a system of variational inequalities in a \(\mathrm{CAT}(0)\) space. Two algorithms are given. Under certain appropriate conditions, we prove that the iterative schemes converge strongly to the unique solution of the hierarchical optimization problem. The result presented in this paper mainly improves and extends the corresponding results of \textit{L. Y. Shi} and \textit{R. D. Chen} [J. Appl. Math. 2012, Article ID 421050, 11 p. (2012; Zbl 1281.47059)], \textit{R. Wangkeeree} and \textit{P. Preechasilp} [J. Inequal. Appl. 2013, Paper No. 93, 15 p. (2013; Zbl 1292.47056)] and others.A Tseng extragradient method for solving variational inequality problems in Banach spaceshttps://zbmath.org/1496.471012022-11-17T18:59:28.764376Z"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.hammad-anuoluwapo|abass.hammed-anuoluwapo"Mebawondu, A. A."https://zbmath.org/authors/?q=ai:mebawondu.akindele-adebayo"Aremu, K. O."https://zbmath.org/authors/?q=ai:aremu.kazeem-olalekanSummary: This paper presents an inertial Tseng extragradient method for approximating a solution of the variational inequality problem. The proposed method uses a single projection onto a half space which can be easily evaluated. The method considered in this paper does not require the knowledge of the Lipschitz constant as it uses variable stepsizes from step to step which are updated over each iteration by a simple calculation. We prove a strong convergence theorem of the sequence generated by this method to a solution of the variational inequality problem in the framework of a 2-uniformly convex Banach space which is also uniformly smooth. Furthermore, we report some numerical experiments to illustrate the performance of this method. Our result extends and unifies corresponding results in this direction in the literature.S-iteration process of Halpern-type for common solutions of nonexpansive mappings and monotone variational inequalitieshttps://zbmath.org/1496.471022022-11-17T18:59:28.764376Z"Sahu, D. R."https://zbmath.org/authors/?q=ai:sahu.daya-ram"Kumar, Ajeet"https://zbmath.org/authors/?q=ai:kumar.ajeet"Wen, Ching-Feng"https://zbmath.org/authors/?q=ai:wen.chingfengSummary: This paper is devoted to the strong convergence of the S-iteration process of Halpern-type for approximating a common element of the set of fixed points of a nonexpansive mapping and the set of common solutions of variational inequality problems formed by two inverse strongly monotone mappings in the framework of Hilbert spaces. We also give some numerical examples in support of our main result.Modified inertial projection and contraction algorithms for solving variational inequality problems with non-Lipschitz continuous operatorshttps://zbmath.org/1496.471042022-11-17T18:59:28.764376Z"Tan, Bing"https://zbmath.org/authors/?q=ai:tan.bing.1"Qin, Xiaolong"https://zbmath.org/authors/?q=ai:qin.xiaolongSummary: In this paper, we present four modified inertial projection and contraction methods to solve the variational inequality problem with a pseudo-monotone and non-Lipschitz continuous operator in real Hilbert spaces. Strong convergence theorems of the proposed algorithms are established without the prior knowledge of the Lipschitz constant of the operator. Several numerical experiments and the applications to optimal control problems are provided to verify the advantages and efficiency of the proposed algorithms.General iterative scheme based on the regularization for solving a constrained convex minimization problemhttps://zbmath.org/1496.471052022-11-17T18:59:28.764376Z"Tian, Ming"https://zbmath.org/authors/?q=ai:tian.mingSummary: It is well known that the regularization method plays an important role in solving a constrained convex minimization problem. In this article, we introduce implicit and explicit iterative schemes based on the regularization for solving a constrained convex minimization problem. We establish results on the strong convergence of the sequences generated by the proposed schemes to a solution of the minimization problem. Such a point is also a solution of a variational inequality. We also apply the algorithm to solve a split feasibility problem.Halpern Tseng's extragradient methods for solving variational inequalities involving semistrictly quasimonotone operatorhttps://zbmath.org/1496.471062022-11-17T18:59:28.764376Z"Wairojjana, Nopparat"https://zbmath.org/authors/?q=ai:wairojjana.nopparat"Pakkaranang, Nuttapol"https://zbmath.org/authors/?q=ai:pakkaranang.nuttapolSummary: In this paper, we study the strong convergence of new methods for solving classical variational inequalities problems involving semistrictly quasimonotone and Lipschitz-continuous operators in a real Hilbert space. Three proposed methods are based on Tseng's extragradient method and use a simple self-adaptive step size rule that is independent of the Lipschitz constant. The step size rule is built around two techniques: the monotone and the non-monotone step size rule. We establish strong convergence theorems for the proposed methods that do not require any additional projections or knowledge of an involved operator's Lipschitz constant. Finally, we present some numerical experiments that demonstrate the efficiency and advantages of the proposed methods.On strong convergence theorems for a viscosity-type Tseng's extragradient methods solving quasimonotone variational inequalitieshttps://zbmath.org/1496.471072022-11-17T18:59:28.764376Z"Wairojjana, Nopparat"https://zbmath.org/authors/?q=ai:wairojjana.nopparat"Pholasa, Nattawut"https://zbmath.org/authors/?q=ai:pholasa.nattawut"Pakkaranang, Nuttapol"https://zbmath.org/authors/?q=ai:pakkaranang.nuttapolSummary: The main goal of this research is to solve variational inequalities involving quasi-monotone operators in infinite-dimensional real Hilbert spaces numerically. The main advantage of these iterative schemes is the ease with which step size rules can be designed based on an operator explanation rather than the Lipschitz constant or another line search method. The proposed iterative schemes use a monotone and non-monotone step size strategy based on mapping (operator) knowledge as a replacement for the Lipschitz constant or another line search method. The strong convergence is demonstrated to correspond well to the proposed methods and to settle certain control specification conditions. Finally, we propose some numerical experiments to assess the effectiveness and influence of iterative methods.Convex contractions of order \(n\) in \(\mathrm{CAT}(0)\) spaceshttps://zbmath.org/1496.471252022-11-17T18:59:28.764376Z"Yildirim, Isa"https://zbmath.org/authors/?q=ai:yildirim.isa"Tekmanli, Yücel"https://zbmath.org/authors/?q=ai:tekmanli.yucel"Khan, Safeer Hussain"https://zbmath.org/authors/?q=ai:khan.safeer-hussainSummary: In this paper, we work on convex contraction of order \(n\). Our first
result in general metric spaces shows that each convex contraction of order \(n\) is a Bessaga mapping. We then turn our attention to \(\mathrm{CAT}(0)\) spaces. We prove a demiclosedness principle for such mappings in this setting. Next, we consider modified Mann iteration process and prove some convergence theorems for fixed points of such mappings in \(\mathrm{CAT}(0)\) spaces. Our results are new in \(\mathrm{CAT}(0)\) setting. Our results remain true in linear spaces like Hilbert and Banach spaces. Finally, we give an example in order to support our main results and to demonstrate the efficiency of modified Mann iteration process.Gamma-convergence of generalized gradient flows with conjugate typehttps://zbmath.org/1496.471262022-11-17T18:59:28.764376Z"Chang, Mao-Sheng"https://zbmath.org/authors/?q=ai:chang.mao-sheng"Liao, Jian-Tong"https://zbmath.org/authors/?q=ai:liao.jian-tongSummary: In this paper we establish the Gamma-convergence of generalized gradient flows with conjugate type. It provides a criteria for obtaining the convergence of generalized gradient flows that correspond to a sort of \(C^1\)-order \(\Gamma \)-convergence of energy functionals and a kind of bounded symmetric positive definite linear operators.Equilibrium problems with complementarity constraints in Banach spaces: stationarity concepts, applications and algorithmshttps://zbmath.org/1496.490012022-11-17T18:59:28.764376Z"Becker, Jan"https://zbmath.org/authors/?q=ai:becker.jan-dirk|becker.jan-michael|becker.jan-steffenSummary: In mathematics, the field of non-cooperative game theory models the competition between several parties, which are called players. Therein, each player tries to reach an individual goal, which is described by an optimization problem. However and in contrast to classical nonlinear programming, there exists a dependency between the players, i.e. the choice of a suitable strategy influences the behavior and the reward of the player's opponents and vice versa. For this reason, a popular solution concept is given by Nash equilibria, which were introduced by John Forbes Nash in his Ph.D. thesis in 1950. In order to prove the existence of a Nash equilibrium, the convexity of the underlying optimization problem is a central requirement. However, this assumption does not hold in general. This thesis is devoted to special equilibrium problems in Banach spaces, which can be described by equilibrium problems with equilibrium/complementarity constraints (EPEC/EPCC). Due to the structure of the underlying feasible set, those games are nonconvex. Motivated by known results with respect to mathematical programs with complementarity constraints, we focus on weaker Nash equilibrium concepts, which can at least be seen as necessary conditions for a Nash equilibrium under suitable assumptions. In the first part of this work, we concentrate on multi-leader multi-follower games, where the participating players are divided hierarchically into leaders and followers, which compete on their particular level with each other. Under suitable assumptions, the solution of the lower level is described by its necessary and sufficient first-order optimality system and can be written as an EPCC. In this context, we first analyze the latter problem in abstract Banach spaces and afterwards, consider the special case of a multi-leader single-follower game (MLFG), where the lower level is given by a quadratic problem in a Hilbert space. For the latter one, we show on the basis of two known penalization techniques that there exist sequences of auxiliary equilibrium problems, which approximate the corresponding EPCC. In the following application that extends known contributions on an optimal control framework of the obstacle problem, we use these auxiliary games and show that both generate sequences, which converge at least to an ϵ-almost C-stationary Nash equilibrium of the original MLFG. The results are analyzed numerically on the basis of a Gauß-Seideltype algorithm and are tested with respect to two examples. The second part is motivated by the work ``A generalized Nash equilibrium approach for optimal control problems of autonomous cars'' by \textit{A. Dreves} and \textit{M. Gerdts} [Optim. Control Appl. Methods 39, No. 1, 326--342 (2018; Zbl 1390.49046)], where a traffic scenario between several intelligent cars is modeled by a dynamic equilibrium problem. Due to the collision avoidance constraint, this game is non-convex. However, we show that it can be written as a generalized Nash equilibrium problem with mixed-integer variables (MINEP), which again is equivalent to an EPCC. In contrast to the first application, we now concentrate on problems in Lebesgue spaces. In the following, we compare known results from abstract Banach spaces and the corresponding ones in Lebesgue spaces. In particular, we show that for general MINEPs all weak Nash equilibrium concepts coincide. Based on these observations, we apply the results to the traffic scenario. In this context, we again use a penalization technique and deduce by the generated sequence of Nash equilibrium problems that we find a sequence of equilibrium points, which converge to an S-stationary Nash equilibrium of MINEP. We end up with a numerical analysis and test the results with two hypothetical traffic scenarios.Nonlinear ultraparabolic equations and variational inequalitieshttps://zbmath.org/1496.490022022-11-17T18:59:28.764376Z"Protsakh, N. P."https://zbmath.org/authors/?q=ai:protsakh.natalia-petrivna|protsakh.nataliya-p"Ptashnyk, B. Ĭ."https://zbmath.org/authors/?q=ai:ptashnyk.bogdan-iosipovich(no abstract)An exact penalty function method for optimal control of a Dubins airplane in the presence of moving obstacleshttps://zbmath.org/1496.490032022-11-17T18:59:28.764376Z"Fathi, Z."https://zbmath.org/authors/?q=ai:fathi.zohreh|fathi.zahra"Bidabad, B."https://zbmath.org/authors/?q=ai:bidabad.behroz"Najafpour, M."https://zbmath.org/authors/?q=ai:najafpour.mIn this paper, the authors investigate the time-optimal control for a Dubins airplane model (it adds altitude to the car model of L.E. Dubins) in the presence of fixed and moving obstacles.
Using an exact penalty function method, the authors show the existence of the control variables for this problem.
The effectiveness of the method is demonstrated using three (toy) examples. In the first example, the moving obstacles are far from the plane's trajectory and consequently have no effect on the trajectory at all. The last two examples deal with moving obstacles disrupting the plane's flight path.
Reviewer: Uwe Prüfert (Freiberg)Existence of minimizers for a generalized liquid drop model with fractional perimeterhttps://zbmath.org/1496.490042022-11-17T18:59:28.764376Z"Novaga, Matteo"https://zbmath.org/authors/?q=ai:novaga.matteo"Onoue, Fumihiko"https://zbmath.org/authors/?q=ai:onoue.fumihikoSummary: We consider the minimization problem of the functional given by the sum of the fractional perimeter and a general Riesz potential, which is one generalization of Gamow's liquid drop model. We first show the existence of minimizers for any volumes if the kernel of the Riesz potential decays faster than that of the fractional perimeter. We also prove the existence of generalized minimizers for any volumes if the kernel of the Riesz potential just vanishes at infinity. Finally, we study the asymptotic behavior of minimizers when the volume goes to infinity and we prove that a sequence of minimizers converges to the Euclidean ball up to translations if the kernel of the Riesz potential decays sufficiently fast.On the applications of a minimax theoremhttps://zbmath.org/1496.490052022-11-17T18:59:28.764376Z"Ricceri, Biagio"https://zbmath.org/authors/?q=ai:ricceri.biagioIn this paper, a minimax theorem is presented and four applications are given: uniquely remotal sets in normed spaces; multiple global minima for the integral functional of the Calculus of Variations; multiple periodic solutions for Lagrangian systems of relativistic oscillators; variational inequalities in balls of Hilbert spaces. A related challenging open problem is also pointed out.
Reviewer: Yang Yang (Wuxi)Self-adaptive inertial projection and contraction algorithm for monotone variational inequalityhttps://zbmath.org/1496.490062022-11-17T18:59:28.764376Z"Gao, Xue"https://zbmath.org/authors/?q=ai:gao.xue"Cai, Xingju"https://zbmath.org/authors/?q=ai:cai.xingju"Wang, Xueye"https://zbmath.org/authors/?q=ai:wang.xueyeApplications of generalized fractional hemivariational inequalities in solid viscoelastic contact mechanicshttps://zbmath.org/1496.490072022-11-17T18:59:28.764376Z"Han, Jiangfeng"https://zbmath.org/authors/?q=ai:han.jiangfeng"Li, Changpin"https://zbmath.org/authors/?q=ai:li.changpin.1|li.changpin"Zeng, Shengda"https://zbmath.org/authors/?q=ai:zeng.shengdaSummary: This paper studies a generalized fractional hemivariational inequality in infinite-dimensional spaces. Under the suitable assumptions, the existence result is delivered by using the temporally semi-discrete scheme and the surjectivity result for multivalued pseudomonotone operator. As an illustrative application, we propose a frictional contact model that describes the quasi-static contact between a viscoelastic body and a solid foundation. The viscoelastic constitutive equation is modeled by fractional Kelvin-Voigt law with \(\psi\)-Caputo derivative, and the frictional contact conditions are expressed as the Clarke subdifferentials of the nonconvex and nonsmooth functionals. Finally, the weak solvability of the mechanical system is obtained by using our abstract mathematical result.Inertial methods for solving system of quasi variational inequalitieshttps://zbmath.org/1496.490082022-11-17T18:59:28.764376Z"Jabeen, Saudia"https://zbmath.org/authors/?q=ai:jabeen.saudia"Noor, Muhammad Aslam"https://zbmath.org/authors/?q=ai:noor.muhammad-aslam"Noor, Khalida Inayat"https://zbmath.org/authors/?q=ai:noor.khalida-inayatSummary: In this paper, we consider a system of quasi variational inequalities involving two arbitrary mappings. It is shown that the system of quasi variational inequalities is equivalent to the fixed point problem using the projection method. We use this alternative formulation to suggest some new inertial projection methods. The convergence criteria of the new methods is analyzed under some appropriate conditions. Several special cases are discussed as applications of the results. It is interesting problem to consider the implementation of the proposed methods with other similar techniques. The concept of this paper may inspire future research in this area. Results obtained in this paper can be viewed as refinement and improvement of previously known results.On optimal solutions of well-posed problems and variational inequalitieshttps://zbmath.org/1496.490092022-11-17T18:59:28.764376Z"Ram, Tirth"https://zbmath.org/authors/?q=ai:ram.tirth"Kim, Jong Kyu"https://zbmath.org/authors/?q=ai:kim.jong-kyu|kim.jongkyu"Kour, Ravdeep"https://zbmath.org/authors/?q=ai:kour.ravdeepThe authors consider two problems connected with variational inequalities. In Section 3 they provide a theorem characterising Tykhonov well-posed problems in locally convex topological vector spaces. Further, in section 4, they study variational inequalities in Hilbert spaces. All results are supplied with simple, illustrative examples.
Reviewer: Michał Bełdziński (Łódź)\(L^\infty\)-truncation of closed differential formshttps://zbmath.org/1496.490102022-11-17T18:59:28.764376Z"Schiffer, Stefan"https://zbmath.org/authors/?q=ai:schiffer.stefanThe paper is devoted to \(L^\infty\)-truncation of closed differential forms. The author proves that for each closed differential form \(u \in L^1(\mathbb{R}^N; (\mathbb{R}^N ) *\wedge\dots\wedge(\mathbb{R}^N ))\), which is almost in \(L^\infty\) in the sense that
\[
\int\limits_ {\{y\in\mathbb{R}^N : |u(y)|\geq L\}} |u(y)|dy < \varepsilon
\]
for some \(L > 0\) and a small \(\varepsilon > 0\), a closed differential form \(v\) can be found, such that the norm \(\|u-v\|_{L^1}\) is again small, and \(v\) is, in addition, in \(L^\infty\) with a bound on its \(L^\infty\) norm depending only on \(N\) and \(L\). In particular, the set \(\{v \neq u\}\) has measure at most \(C L^{-1}\varepsilon\). To prove this a special lemma is formulated which can be seen as a natural analogue of Lipschitz continuity on the set where the maximal function is small and the geometric Whitney extension theorem are established. The applications of the results to \(A\)-quasiconvex hulls and \(A\)-Young measures are also discussed.
Reviewer: Aygul Manapova (Ufa)Correction to: ``Brezis pseudomonotone bifunctions and quasi equilibrium problems via penalization''https://zbmath.org/1496.490112022-11-17T18:59:28.764376Z"Bianchi, M."https://zbmath.org/authors/?q=ai:bianchi.monica"Kassay, G."https://zbmath.org/authors/?q=ai:kassay.gabor"Pini, R."https://zbmath.org/authors/?q=ai:pini.ritaCorrection to the authors' paper [ibid. 82, No. 3, 483--498 (2022; Zbl 1484.49032)].Polyhedral optimization of second-order discrete and differential inclusions with delayhttps://zbmath.org/1496.490122022-11-17T18:59:28.764376Z"Sağlam, Sevilay Demir"https://zbmath.org/authors/?q=ai:saglam.sevilay-demir"Mahmudov, Elimhan N."https://zbmath.org/authors/?q=ai:mahmudov.elimhan-nadirSummary: The present paper studies the optimal control theory of second-order polyhedral delay discrete and delay differential inclusions with state constraints. We formulate the conditions of optimality for the problems with the second-order polyhedral delay discrete (\(PD_d\)) and the delay differential (\(PC_d\)) in terms of the Euler-Lagrange inclusions and the distinctive ``transversality'' conditions. Moreover, some linear control problem with second-order delay differential inclusions is given to illustrate the effectiveness and usefulness of the main theoretic results.Asymptotic properties of an optimal principal eigenvalue with spherical weight and Dirichlet boundary conditionshttps://zbmath.org/1496.490132022-11-17T18:59:28.764376Z"Ferreri, Lorenzo"https://zbmath.org/authors/?q=ai:ferreri.lorenzo"Verzini, Gianmaria"https://zbmath.org/authors/?q=ai:verzini.gianmariaSummary: We consider a weighted eigenvalue problem for the Dirichlet laplacian in a smooth bounded domain \(\Omega \subset \mathbb{R}^N\), where the bang-bang weight equals a positive constant \(\overline{m}\) on a ball \(B \subset \Omega\) and a negative constant \(- \underline{m}\) on \(\Omega \backslash B\). The corresponding positive principal eigenvalue provides a threshold to detect persistence/extinction of a species whose evolution is described by the heterogeneous Fisher-KPP equation in population dynamics. In particular, we study the minimization of such eigenvalue with respect to the position of \(B\) in \(\Omega \). We provide sharp asymptotic expansions of the optimal eigenpair in the singularly perturbed regime in which the volume of \(B\) vanishes. We deduce that, up to subsequences, the optimal ball concentrates at a point maximizing the distance from \(\partial \Omega \).On stability of periodic solutions of a ``sweeping'' processhttps://zbmath.org/1496.490142022-11-17T18:59:28.764376Z"Voskovskaya, Natalia Igorevna"https://zbmath.org/authors/?q=ai:voskovskaya.natalia-igorevnaSummary: This paper is dedicated to the so-called ``sweeping'' process that plays an important role in elastoplasticity, quasistatics and dynamics. In this paper we consider the following ``sweeping'' process: an ellipse described by the equation \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\), \(a>b\), moves by a periodic law along its major axis, which lays at abscissa axis. The ellipse contains a point inside or on its boundary and this point follows its movement. The purpose of the paper is to study the stability of solutions of this ``sweeping'' process. \par We prove that unique periodic solution of this ``sweeping'' process is globally stable.Optimal control for uncertain random singular systems with multiple time-delayshttps://zbmath.org/1496.490152022-11-17T18:59:28.764376Z"Chen, Xin"https://zbmath.org/authors/?q=ai:chen.xin|chen.xin.1"Zhu, Yuanguo"https://zbmath.org/authors/?q=ai:zhu.yuanguoSummary: Chance theory provides us a useful tool to solve an optimal control problem with indeterminacy composing of both uncertainty and randomness. Based on chance theory, this paper studies an optimal control for uncertain random singular systems with multiple time-delays. First, an uncertain random singular system with multiple time-delays is introduced, and then the corresponding optimal control problem is established. The equivalent relationship between this problem and the optimal control problem for standard uncertain random systems is derived. Then the appropriate recurrence equations are proposed according to the dynamic programming method. Furthermore, two kinds of optimal control problems are discussed. The optimal control inputs and respective optimal values of the problems are provided via the solvability of the obtained equations. Finally, a numerical example is presented to show the effectiveness of our theoretical results.A shape Newton scheme for deforming shells with application to capillary bridgeshttps://zbmath.org/1496.490162022-11-17T18:59:28.764376Z"Schmidt, Stephan"https://zbmath.org/authors/?q=ai:schmidt.stephan"Gräßer, Melanie"https://zbmath.org/authors/?q=ai:grasser.melanie"Schmid, Hans-Joachim"https://zbmath.org/authors/?q=ai:schmid.hans-joachimAn iterative method for solving the multiple-sets split variational inequality problemhttps://zbmath.org/1496.490172022-11-17T18:59:28.764376Z"Cuong, Tran Luu"https://zbmath.org/authors/?q=ai:cuong.tran-luu"Anh, Tran Viet"https://zbmath.org/authors/?q=ai:anh.tran-vietIn the paper an iterative algorithm for finding minimum-norm solutions of the multiple-sets split variational inequality problem in real Hilbert spaces is presented. Under the condition that the mappings are monotone and Lipschitz continuous, the strong convergence of the iterative sequence generated by the algorithm method is proved. Some applications to the multiple-sets split feasibility problem and the split variational inequality problem are considered. A simple numerical example is performed to illustrate the performance of the proposed algorithm.
Reviewer: Aygul Manapova (Ufa)A suboptimal control of linear time-delay problems via dynamic programminghttps://zbmath.org/1496.490182022-11-17T18:59:28.764376Z"Gooran Orimi, Atefeh"https://zbmath.org/authors/?q=ai:gooran-orimi.atefeh"Effati, Sohrab"https://zbmath.org/authors/?q=ai:effati.sohrab"Farahi, Mohammad Hadi"https://zbmath.org/authors/?q=ai:farahi.mohammad-hadiSummary: We study a class of infinite horizon optimal control problems with a state delay, and investigate the dynamic programming approach which leverages the sufficient optimality conditions and provides a closed-loop solution. Importantly, the well-known Lyapunov-Krasovskii functional is applied to relate the solution of the problem to the solution of a set of three Riccati-type matrix differential equations. We then present an analytic-based approach to solve the resultant equations and subsequently provide a suboptimal closed-loop solution for the considered problem. We prove the uniform convergence of the proposed approach and show that the presented closed-loop system is asymptotically stable in the Lyapunov sense. Furthermore, the observability of the linear time-delay system is discussed and proved. Finally, numerical examples illustrate the efficiency of the proposed method.Mean field verification theoremhttps://zbmath.org/1496.490192022-11-17T18:59:28.764376Z"Bensoussan, Alain"https://zbmath.org/authors/?q=ai:bensoussan.alain"Hoe, SingRu (Celine)"https://zbmath.org/authors/?q=ai:hoe.singru"Kim, Joohyun"https://zbmath.org/authors/?q=ai:kim.joohyun"Yan, Zhongfeng"https://zbmath.org/authors/?q=ai:yan.zhongfengThis paper describes the main ideas of the verification therorem for mean field control problems. More exactly, the considered problem is the following: find a control in a feedback form for a stochastic differential equation maximizing a given payoff function. Then, the authors extend Bellman equation of stochastic control to this situation and obtain a verification theorem for a specific feedback.
Reviewer: Nicolae Cîndea (Aubière)Approximation and mean field control of systems of large populationshttps://zbmath.org/1496.490202022-11-17T18:59:28.764376Z"Higuera-Chan, Carmen G."https://zbmath.org/authors/?q=ai:higuera-chan.carmen-gSummary: We deal with a class of discrete-time stochastic controlled systems composed by a large population of \(N\) interacting individuals. Given that \(N\) is large and the cost function is possibly unbounded, the problem is studied by means of a limit model \(\mathcal{M}\), known as the mean field model, which is obtained as limit as \(N \rightarrow \infty\) of the model \(\mathcal{M}_N\) corresponding to the system of \(N\) individuals in combination with an approximate algorithm for the cost function.
For the entire collection see [Zbl 1478.60006].Simultaneous shape and mesh quality optimization using pre-shape calculushttps://zbmath.org/1496.490212022-11-17T18:59:28.764376Z"Luft, Daniel"https://zbmath.org/authors/?q=ai:luft.daniel"Schulz, Volker"https://zbmath.org/authors/?q=ai:schulz.volker-hSummary: Computational meshes arising from shape optimization routines commonly suffer from decrease of mesh quality or even destruction of the mesh. In this work, we provide an approach to regularize general shape optimization problems to increase both shape and volume mesh quality. For this, we employ pre-shape calculus as established in [\textit{D. Luft} and \textit{V. Schulz}, Control Cybern. 50, No. 3, 263--301 (2021; Zbl 1495.49028)]. Existence of regularized solutions is guaranteed. Further, consistency of modified pre-shape gradient systems is established. We present pre-shape gradient system modifications, which permit simultaneous shape optimization with mesh quality improvement. Optimal shapes to the original problem are left invariant under regularization. The computational burden of our approach is limited, since additional solution of possibly larger (non-)linear systems for regularized shape gradients is not necessary. We implement and compare pre-shape gradient regularization approaches for a 2D problem, which is prone to mesh degeneration. As our approach does not depend on the choice of metrics representing shape gradients, we employ and compare several different metrics.Gaffney-Friedrichs inequality for differential forms on Heisenberg groupshttps://zbmath.org/1496.490222022-11-17T18:59:28.764376Z"Franchi, Bruno"https://zbmath.org/authors/?q=ai:franchi.bruno"Montefalcone, Francescopaolo"https://zbmath.org/authors/?q=ai:montefalcone.francescopaolo"Serra, Elena"https://zbmath.org/authors/?q=ai:serra.elenaSummary: In this paper, we will prove several generalized versions, dependent on different boundary conditions, of the classical Gaffney-Friedrichs inequality for differential forms on Heisenberg groups. In the first part of the paper, we will consider horizontal differential forms and the horizontal differential. In the second part, we shall prove the counterpart of these results in the context of Rumin's complex.Isoperimetric clusters in homogeneous spaces via concentration compactnesshttps://zbmath.org/1496.490232022-11-17T18:59:28.764376Z"Novaga, Matteo"https://zbmath.org/authors/?q=ai:novaga.matteo"Paolini, Emanuele"https://zbmath.org/authors/?q=ai:paolini.emanuele"Stepanov, Eugene"https://zbmath.org/authors/?q=ai:stepanov.eugene"Tortorelli, Vincenzo Maria"https://zbmath.org/authors/?q=ai:tortorelli.vincenzo-mariaSummary: We show the existence of generalized clusters of a finite or even infinite number of sets, with minimal total perimeter and given total masses, in metric measure spaces homogeneous with respect to a group acting by measure preserving homeomorphisms, for a quite wide range of perimeter functionals. Such generalized clusters are a natural ``relaxed'' version of a cluster and can be thought of as ``albums'' with possibly infinite pages, having a minimal cluster drawn on each page, the total perimeter and the vector of masses being calculated by summation over all pages, the total perimeter being minimal among all generalized clusters with the same masses. The examples include any anisotropic perimeter in a Euclidean space, as well as a hyperbolic plane with the Riemannian perimeter and Heisenberg groups with a canonical left invariant perimeter or its equivalent versions.A Steiner inequality for the anisotropic perimeterhttps://zbmath.org/1496.490242022-11-17T18:59:28.764376Z"Dai, Jin"https://zbmath.org/authors/?q=ai:dai.jinThis paper proves a Steiner inequality for the anisotropic perimeter. As a consequence of this Steiner inequality, the author obtains a more direct proof for the Wulff inequality.
This article is structured in four sections. The first section introduces the main result briefly recalling the state of art of this subject. Section 2 gathers the definitions and results needed for the proof of the main results which makes the object of Sections 3 and 4.
Reviewer: Nicolae Cîndea (Aubière)Variational approach to regularity of optimal transport maps: general cost functionshttps://zbmath.org/1496.490252022-11-17T18:59:28.764376Z"Otto, Felix"https://zbmath.org/authors/?q=ai:otto.felix"Prod'homme, Maxime"https://zbmath.org/authors/?q=ai:prodhomme.maxime"Ried, Tobias"https://zbmath.org/authors/?q=ai:ried.tobiasThe paper continues the line of research started with [\textit{M. Goldman} and \textit{F. Otto}, Ann. Sci. Éc. Norm. Supér. (4) 53, No. 5, 1209--1233 (2020; Zbl 1465.35263)] and [\textit{M. Goldman} et al., Commun. Pure Appl. Math. 74, No. 12, 2483--2560 (2021; Zbl 1480.35082)]. The aim is the study of the regularity theory for optimal transport maps with a purely variational approach.
Here the authors deal with general cost functions and Hölder continuous densities, obtaining a slightly more quantitative result than the one in the celebrated paper [\textit{G. De Philippis} and \textit{A. Figalli}, Publ. Math., Inst. Hautes Étud. Sci. 121, 81--112 (2015; Zbl 1325.49051)]. Besides the differences in the statement of the main result compared to the one of De Philippis and Figalli, the interest is in the different approach. Indeed, the authors do not rely on the regularity theory for the Monge-Ampère equation and use arguments similar to De Giorgi's strategy for the \(\varepsilon\)-regularity of minimal surfaces.
The result can be also applied to the study of the optimal transport problem on Riemannian manifolds with cost given by the square of the Riemannian distance.
Reviewer: Nicolò De Ponti (Trieste)A unified approach to collectively maximal elements in abstract convex spaceshttps://zbmath.org/1496.520022022-11-17T18:59:28.764376Z"Park, Sehie"https://zbmath.org/authors/?q=ai:park.sehieThe author establishes a very KKM type theorem in abstract convex spaces from which he then obtains an abstract collectively maximal element theorem. Finally, the author shows that a large number of previous theorems on the existence of maximal element and of equilibrium can be derived from his result.
Reviewer: Mircea Balaj (Oradea)A Möbius invariant discretization of O'Hara's Möbius energyhttps://zbmath.org/1496.570042022-11-17T18:59:28.764376Z"Blatt, Simon"https://zbmath.org/authors/?q=ai:blatt.simon|blatt.simon.1"Ishizeki, Aya"https://zbmath.org/authors/?q=ai:ishizeki.aya"Nagasawa, Takeyuki"https://zbmath.org/authors/?q=ai:nagasawa.takeyukiThe authors introduce a new discretization of O'Hara's Möbius energy. In contrast to the known discretizations of \textit{J. K. Simon} [J. Knot Theory Ramifications 3, No. 3, 299--320 (1994; Zbl 0841.57017)] and of \textit{D. Kim} and \textit{R. Kusner} [Exp. Math. 2, No. 1, 1--9 (1993; Zbl 0818.57007)] the new discretization is invariant under Möbius transformations of the surrounding space. Moreover, this energy is minimized by polygons with vertices on a circle. The starting point for this new discretization is the so-called cosine formula of Doyle and Schramm. In addition, the authors then show \(\Gamma\)-convergence of the discretized energy to the Möbius energy provided that the fineness of the polygons is going to 0. (Here a map \(p : \mathbb{R/Z}\to \mathbb{R}^n\) a closed polygon with the \(m\) vertices \(p(\theta_i)\in\mathbb{R}^n\), \(i=1,\ldots,m\) if there are points \(\theta_i \in[0,1)\), \(\theta_1 <\theta_2 <\dots<\theta_m\) such that \(p\) is linear between two neighboring points \(\theta_i\) and \(\theta_{i+1}\) with \(\theta_{m+1}=\theta_1\). The fineness is then defined as \(\max|\theta_{i+1}-\theta_i|\).)
Reviewer: Claus Ernst (Bowling Green)A variational principle for Kaluza-Klein types theorieshttps://zbmath.org/1496.580032022-11-17T18:59:28.764376Z"Hélein, Frédéric"https://zbmath.org/authors/?q=ai:helein.fredericSummary: For any positive integer \(n\) and any Lie group \(\mathfrak{G}\), given a definite symmetric bilinear form on \(\mathbb{R}^n\) and an Ad-invariant scalar product on the Lie algebra of \(\mathfrak{G} \), we construct a variational problem on fields defined on an arbitrary oriented \((n + \dim\mathfrak{G})\)-dimensional manifold \(\mathcal{Y}\). We show that, if \(\mathfrak{G}\) is compact and simply connected, any global solution of the Euler-Lagrange equations leads, through a spontaneous symmetry breaking, to identify \(\mathcal{Y}\) with the total space of a principal bundle over an n-dimensional manifold \(\mathcal{X} \). Moreover \(\mathcal{X}\) is then endowed with a (pseudo-)Riemannian metric and a connection which are solutions of the Einstein-Yang-Mills system of equations with a cosmological constant.Reduced cost numerical methods of sixth-order convergence for systems of nonlinear modelshttps://zbmath.org/1496.650652022-11-17T18:59:28.764376Z"Singh, Harmandeep"https://zbmath.org/authors/?q=ai:singh.harmandeep"Sharma, Janak Raj"https://zbmath.org/authors/?q=ai:sharma.janak-rajSummary: The objective of present study is to develop the iterative methods with higher convergence order but keeping the mathematical computations as small as possible. With this objective, two multi-step sixth order methods have been designed by utilizing only two Jacobian matrices and single matrix inversion apart from three functional evaluations. The techniques with these characteristics are rarely found in the literature. Comparing in the context of computational efficiency, both of the developed methods are exceptional, and outperform the existing methods. Numerical performance is analyzed by executing the experimentation on the selected nonlinear problems. Outcomes of the analysis are remarkable and significantly favor the new methods as compared to their existing counterparts, typically for large scale systems.A new family of hybrid three-term conjugate gradient methods with applications in image restorationhttps://zbmath.org/1496.650762022-11-17T18:59:28.764376Z"Jiang, Xianzhen"https://zbmath.org/authors/?q=ai:jiang.xianzhen"Liao, Wei"https://zbmath.org/authors/?q=ai:liao.wei"Yin, Jianghua"https://zbmath.org/authors/?q=ai:yin.jianghua"Jian, Jinbao"https://zbmath.org/authors/?q=ai:jian.jinbaoSummary: In this paper, based on the hybrid conjugate gradient method and the convex combination technique, a new family of hybrid three-term conjugate gradient methods are proposed for solving unconstrained optimization. The conjugate parameter in the search direction is a hybrid of Dai-Yuan conjugate parameter and any one. The search direction then is the sum of the negative gradient direction and a convex combination in relation to the last search direction and the gradient at the previous iteration. Without choosing any specific conjugate parameters, we show that the search direction generated by the family always possesses the descent property independent of line search technique, and that it is globally convergent under usual assumptions and the weak Wolfe line search. To verify the effectiveness of the presented family, we further design a specific conjugate parameter, and perform medium-large-scale numerical experiments for smooth unconstrained optimization and image restoration problems. The numerical results show the encouraging efficiency and applicability of the proposed methods even compared with the state-of-the-art methods.An adaptive mesh refinement method for indirectly solving optimal control problemshttps://zbmath.org/1496.650782022-11-17T18:59:28.764376Z"Yang, Chaoyi"https://zbmath.org/authors/?q=ai:yang.chaoyi"Fabien, Brian C."https://zbmath.org/authors/?q=ai:fabien.brian-cSummary: The indirect solution of optimal control problems (OCPs) with inequality constraints and parameters is obtained by solving the two-point boundary value problem (BVP) involving index-1 differential-algebraic equations (DAEs) associated with its first-order optimality conditions. This paper introduces an adaptive mesh refinement method based on a collocation method for solving the index-1 BVP-DAEs. The paper first derives a method to estimate the relative error between the numerical solution and the exact solution. The relative error estimate is then used to guide the mesh refinement process. The mesh size is increased when the estimated error within a mesh interval is beyond the numerical tolerance by either increasing the order of the approximating polynomial or dividing the interval into multiple subintervals. In the mesh interval where the error tolerance has been met, the mesh size is reduced by either decreasing the degree of the approximating polynomial or merging adjacent mesh intervals. An efficient parallel implementation of the method is implemented using Python and CUDA. The paper presents three examples which show that the approach is more computationally efficient and robust when compared with fixed-order methods.A second-order dynamical system for equilibrium problemshttps://zbmath.org/1496.650812022-11-17T18:59:28.764376Z"Le Van Vinh"https://zbmath.org/authors/?q=ai:le-van-vinh."Van Nam Tran"https://zbmath.org/authors/?q=ai:van-nam-tran."Phan Tu Vuong"https://zbmath.org/authors/?q=ai:vuong.phan-tuSummary: We consider a second-order dynamical system for solving equilibrium problems in Hilbert spaces. Under mild conditions, we prove existence and uniqueness of strong global solution of the proposed dynamical system. We establish the exponential convergence of trajectories under strong pseudo-monotonicity and Lipschitz-type conditions. We then investigate a discrete version of the second-order dynamical system, which leads to a fixed point-type algorithm with inertial effect and relaxation. The linear convergence of this algorithm is established under suitable conditions on parameters. Finally, some numerical experiments are reported confirming the theoretical results.A numerical solution for fractional linear quadratic optimal control problems via shifted Legendre polynomialshttps://zbmath.org/1496.650972022-11-17T18:59:28.764376Z"Nezhadhosein, Saeed"https://zbmath.org/authors/?q=ai:nezhadhosein.saeed"Ghanbari, Reza"https://zbmath.org/authors/?q=ai:ghanbari.reza"Ghorbani-Moghadam, Khatere"https://zbmath.org/authors/?q=ai:ghorbani-moghadam.khatere(no abstract)An optimal mass transport method for random genetic drifthttps://zbmath.org/1496.651092022-11-17T18:59:28.764376Z"Carrillo, José A."https://zbmath.org/authors/?q=ai:carrillo.jose-antonio"Chen, Lin"https://zbmath.org/authors/?q=ai:chen.lin.2|chen.lin.5|chen.lin.1|chen.lin.3|chen.lin|chen.lin.6|chen.lin.4"Wang, Qi"https://zbmath.org/authors/?q=ai:wang.qiA second-order accurate, energy stable numerical scheme for the one-dimensional porous medium equation by an energetic variational approachhttps://zbmath.org/1496.651102022-11-17T18:59:28.764376Z"Duan, Chenghua"https://zbmath.org/authors/?q=ai:duan.chenghua"Chen, Wenbin"https://zbmath.org/authors/?q=ai:chen.wenbin"Liu, Chun"https://zbmath.org/authors/?q=ai:liu.chun"Wang, Cheng"https://zbmath.org/authors/?q=ai:wang.cheng.1"Yue, Xingye"https://zbmath.org/authors/?q=ai:yue.xingyeSummary: The porous medium equation (PME) is a typical nonlinear degenerate parabolic equation. An energetic variational approach (EVA) provides many insights to such a physical model, in which the trajectory equation can be obtained, based on different dissipative energy laws. In this article, we propose and analyze a second-order accurate-in-time numerical scheme for the PME in the EVA approach. A modified Crank-Nicolson temporal discretization is applied, combined with the finite difference over a uniform spatial mesh. Such a numerical scheme is highly nonlinear, and it is proved to be uniquely solvable on an admissible convex set, in which the convexity of the nonlinear implicit terms will play an important role. Subsequently, the energy dissipation property is established, with careful summation by parts formulas applied in the spatial discretization. More importantly, an optimal rate convergence analysis is provided in this work, in which many highly non-standard estimates have to be involved, due to the nonlinear parabolic coefficients. The higher order asymptotic expansion (up to fourth-order temporal and spatial accuracy), the rough error estimate (to establish the \(W^{1,\infty}_h\) bound for the numerical variable), and the refined error estimate have to be carried out to accomplish such a convergence result. In our knowledge, it will be the first work to combine three theoretical properties for a second order accurate numerical scheme to the PME in the EVA approach: unique solvability, energy stability and optimal rate convergence analysis. A few numerical results are also presented in this article, which demonstrate the robustness of the proposed numerical scheme.An asymptotical regularization with convex constraints for inverse problemshttps://zbmath.org/1496.651492022-11-17T18:59:28.764376Z"Zhong, Min"https://zbmath.org/authors/?q=ai:zhong.min"Wang, Wei"https://zbmath.org/authors/?q=ai:wang.wei.33"Tong, Shanshan"https://zbmath.org/authors/?q=ai:tong.shanshanThe subject of this paper is the nonlinear equation \( F(x) = y \), where \( F: \mathcal{D}(F) \subset \mathcal{X} \to \mathcal{Y} \) is a nonlinear, Fréchet differentiable operator between two infinite-dimensional Hilbert spaces \( \mathcal{X} \) and \( \mathcal{Y} \). Instead of exact data \( y \), only perturbed data \( y^\delta \in \mathcal{Y} \) satisfying \( \Vert y^\delta - y \Vert \le \delta \) are available. Let \( \Theta: \mathcal{X} \to (-\infty,\infty] \) be a proper, lower semicontinuous, uniformly convex functional which can be utilized to identify different features of the solution.
The authors consider a linearized asymptotical regularization, with a discrepancy principle as stopping criterion. In what follows, we give some details. For this purpose, let some initial time \( T_0^\delta = T_0 \), an initial guess \( \xi^\delta(T_0^\delta) = \xi_0 \in \mathcal{X} \) and \( x^\delta(T_0^\delta) = \nabla \Theta^*(\xi_0) \) are chosen, where \( \Theta^*(\xi) = \sup_{x\in\mathcal{X}} \{ \langle \xi,x \rangle - \Theta(x) \}\), \(\xi \in \mathcal{X} \) denotes the Legendre-Fenchel conjugate of \( \Theta \). For some time \( T_n^\delta > 0 \) and a pair \( (\xi^\delta(T_n^\delta), x^\delta(T_n^\delta)) \) obtained by the \(n\)th step (\(n\ge0\)), the next asymptotical regularization step is given by the coupled linearized equation
\[
\frac{d\xi^\delta(t) }{dt} = F^\prime(x^\delta(T_n^\delta))^* s_n^\delta(t), \quad x^\delta(t) = \nabla \Theta^*(\xi^\delta(t)), \quad t > T_n^\delta,
\]
where \( s_n^\delta(t) = y^\delta - F(x^\delta(T_n^\delta)) - F^\prime(x^\delta(T_n^\delta))(x^\delta(t)-x^\delta(T_n^\delta)) \) denotes a linearization. This coupled system is solved for \( T_n^\delta < t < T_{n+1}^\delta \) where the final time \( T_{n+1}^\delta > T_{n}^\delta \) is determined by the conditions \( \Vert s_n^\delta(t) \Vert > \gamma \Vert F(x^\delta(T_n^\delta)) - y^\delta \Vert \) for \( T_n^\delta \le t < T_{n+1}^\delta \) and \( \Vert s_n^\delta(t) \Vert = \gamma \Vert F(x^\delta(T_n^\delta)) - y^\delta \Vert \) for \( t = T_{n+1}^\delta \). This procedure stops with \( n = n_\delta \) if \( \Vert F(x^\delta(T_{n}^\delta))-y^\delta \Vert \le \tau \delta \) is satisfied for the first time. Here, \( 0 < \gamma < 1 \) and \( \tau > 1 \) denote suitable constants.
Convergence in the case of exact data (\(\delta = 0\), no stopping rule) is established. Another issue of the paper is stability of the method with respect to perturbations of \( y \), for \( n \) fixed. Finally, convergence rates are obtained under Hölder continuity of the inverse mapping \( F^{-1} \). Convergence and stability are considered both with respect to norm and the Bregman distance. The paper concludes with some numerical examples which are based on Runge-Kutta discretization.
Reviewer: Robert Plato (Siegen)Control and numerical approximation of fractional diffusion equationshttps://zbmath.org/1496.651552022-11-17T18:59:28.764376Z"Biccari, Umberto"https://zbmath.org/authors/?q=ai:biccari.umberto"Warma, Mahamadi"https://zbmath.org/authors/?q=ai:warma.mahamadi"Zuazua, Enrique"https://zbmath.org/authors/?q=ai:zuazua.enriqueSummary: The aim of this chapter is to give a broad panorama of the control properties of fractional diffusive models from a numerical analysis and simulation perspective. We do this by surveying several research results we obtained in the last years, focusing in particular on the numerical computation of controls, though not forgetting to recall other relevant contributions which can be currently found in the literature of this prolific field. Our reference model will be a non-local diffusive dynamics driven by the fractional Laplacian on a bounded domain \(\Omega\). The starting point of our analysis will be a Finite Element approximation for the associated elliptic model in one and two space-dimensions, for which we also present error estimates and convergence rates in the \(L^2\) and energy norm. Secondly, we will address two specific control scenarios: firstly, we consider the standard interior control problem, in which the control is acting from a small subset \(\omega \subset \Omega\). Secondly, we move our attention to the exterior control problem, in which the control region \(\mathcal{O} \subset \Omega^c\) is located outside \(\Omega\). This exterior control notion extends boundary control to the fractional framework, in which the non-local nature of the models does not allow for controls supported on \(\partial \Omega\). We will conclude by discussing the interesting problem of simultaneous control, in which we consider families of parameter-dependent fractional heat equations and we aim at designing a unique control function capable of steering all the different realizations of the model to the same target configuration. In this framework, we will see how the employment of stochastic optimization techniques may help in alleviating the computational burden for the approximation of simultaneous controls. Our discussion is complemented by several open problems related with fractional models which are currently unsolved and may be of interest for future investigation.
For the entire collection see [Zbl 1492.49003].On the quasiconvex hull for a three-well problem in two dimensional linear elasticityhttps://zbmath.org/1496.740282022-11-17T18:59:28.764376Z"Capella, A."https://zbmath.org/authors/?q=ai:capella.antonio"Morales, L."https://zbmath.org/authors/?q=ai:morales.linda|morales.l-e-mendoza|morales.lino-garcia|morales.lluvia|morales.luis-b|morales.lola|morales.lisa|morales.leopoldo|morales.luiza-aSummary: We provide quantitative inner and outer bounds for the symmetric quasiconvex hull \(Q^e(\mathcal{U})\) on linear strains generated by three-well sets \(\mathcal{U}\) in \(\mathbb{R}^{2\times 2}_{sym}\). In our study, we consider all possible compatible configurations for three wells and prove that if there exist two matrices in \(\mathcal{U}\) that are rank-one compatible then \(Q^e(\mathcal{U})\) coincides with its symmetric lamination convex hull \(L^e(\mathcal{U})\). We complete this result by providing an explicit characterization of \(L^e(\mathcal{U})\) in terms of the wells in \(\mathcal{U}\). Finally, we discuss the optimality of our outer bound and its relationship with quadratic polyconvex functions.Algorithmization in structural optimizationhttps://zbmath.org/1496.741072022-11-17T18:59:28.764376Z"Nazirov, Sh. A."https://zbmath.org/authors/?q=ai:nazirov.shodmankula-abdirozikovich"Saidov, U. M."https://zbmath.org/authors/?q=ai:saidov.u-m(no abstract)The calibration method for the thermal insulation functionalhttps://zbmath.org/1496.800032022-11-17T18:59:28.764376Z"Labourie, C."https://zbmath.org/authors/?q=ai:labourie.camille"Milakis, E."https://zbmath.org/authors/?q=ai:milakis.emmanouilSummary: We provide minimality criteria by construction of calibrations for functionals arising in the theory of thermal insulation.The factorization method for inverse scattering by a two-layered cavity with conductive boundary conditionhttps://zbmath.org/1496.810922022-11-17T18:59:28.764376Z"Ye, Jianguo"https://zbmath.org/authors/?q=ai:ye.jianguo"Yan, Guozheng"https://zbmath.org/authors/?q=ai:yan.guozhengSummary: In this paper we consider the inverse scattering problem of determining the shape of a two-layered cavity with conductive boundary condition from sources and measurements placed on a curve inside the cavity. First, we show the well-posedness of the direct scattering problem by using the boundary integral equation method. Then, we prove that the factorization method can be applied to reconstruct the interface of the two-layered cavity from near-field data. Some numerical experiments are also presented to demonstrate the feasibility and effectiveness of the factorization method.Transition pathways in cylinder-gyroid interfacehttps://zbmath.org/1496.820272022-11-17T18:59:28.764376Z"Yao, Xiaomei"https://zbmath.org/authors/?q=ai:yao.xiaomei"Xu, Jie"https://zbmath.org/authors/?q=ai:xu.jie"Zhang, Lei"https://zbmath.org/authors/?q=ai:zhang.lei.4Summary: When two distinct ordered phases contact, the interface may exhibit rich and fascinating structures. Focusing on the Cylinder-Gyroid interface system, transition pathways connecting various interface morphologies are studied armed with the Landau-Brazovskii model. Specifically, minimum energy paths are obtained by computing transition states with the saddle dynamics. We present four primary transition pathways connecting different local minima, representing four different mechanisms of the formation of the Cylinder-Gyroid interface. The connection of Cylinder and Gyroid can be either direct or indirect via Fddd with three different orientations. Under different displacements, each of the four pathways may have the lowest energy.\texttt{Tenscalc}: a toolbox to generate fast code to solve nonlinear constrained minimizations and compute Nash equilibriahttps://zbmath.org/1496.900042022-11-17T18:59:28.764376Z"Hespanha, João P."https://zbmath.org/authors/?q=ai:hespanha.joao-pedroSummary: We describe the toolbox \texttt{Tenscalc} that generates specialized C-code to solve nonlinear constrained optimizations and to compute Nash equilibria. \texttt{Tenscalc} is aimed at scenarios where one needs to solve very fast a large number of optimizations that are structurally similar. This is common in applications where the optimizations depend on measured data and one wants to compute optima for large or evolving datasets, e.g., in robust estimation and classification, maximum likelihood estimation, model predictive control (MPC), moving horizon estimation (MHE), and combined MPC-MHE (which requires the computation of a saddle-point equilibria). \texttt{Tenscalc} is mostly aimed at generating solvers for optimizations with up to a few thousands of optimization variables/constraints and solve times up to a few milliseconds. The speed achieved by the solver arises from a combination of features: reuse of intermediate computations across and within iterations of the solver, detection and exploitation of matrix sparsity, avoidance of run-time memory allocation and garbage collection, and reliance on flat code that improves the efficiency of the microprocessor pipelining and caching. All these features have been automated and embedded into the code generation process. We include a few representative examples to illustrate how the speed and memory footprint of the solver scale with the size of the problem.Advances in optimization and applications. 12th international conference, OPTIMA 2021, Petrovac, Montenegro, September 27 -- October 1, 2021. Revised selected papershttps://zbmath.org/1496.900072022-11-17T18:59:28.764376ZPublisher's description: This book constitutes the refereed proceedings of the 12th International Conference on Optimization and Applications, OPTIMA 2021, held in Petrovac, Montenegro, in September -- October 2021. Due to the COVID-19 pandemic the conference was partially held online.
The 19 revised full papers presented were carefully reviewed and selected from 38 submissions. The papers are organized in topical sections on mathematical programming; global optimization; stochastic optimization; optimal control; mathematical economics; optimization in data analysis; applications.
The articles of this volume will be reviewed individually. For the preceding conference see [Zbl 07347442].
Indexed articles:
\textit{Birjukov, A.; Chernov, A.}, On numerical estimates of errors in solving convex optimization problems, 3-18 [Zbl 07624786]
\textit{Kuruzov, Ilya A.; Stonyakin, Fedor S.}, Sequential subspace optimization for quasar-convex optimization problems with inexact gradient, 19-33 [Zbl 07624787]
\textit{Barkalov, Konstantin; Usova, Marina}, A search algorithm for the global extremum of a discontinuous function, 37-49 [Zbl 07624788]
\textit{Vedel, Yana; Semenov, Vladimir; Denisov, Sergey}, A novel algorithm with self-adaptive technique for solving variational inequalities in Banach spaces, 50-64 [Zbl 07624789]
\textit{Buldaev, Alexander; Kazmin, Ivan}, On one method of optimization of quantum systems based on the search for fixed points, 67-81 [Zbl 07624790]
\textit{Pasechnyuk, Dmitry; Dvurechensky, Pavel; Omelchenko, Sergey; Gasnikov, Alexander}, Stochastic optimization for dynamic pricing, 82-94 [Zbl 07624791]
\textit{Safin, Kamil; Dvurechensky, Pavel; Gasnikov, Alexander}, Adaptive gradient-free method for stochastic optimization, 95-108 [Zbl 07624792]
\textit{Aida-Zade, K. R.; Hashimov, V. A.}, Synthesis of power and movement control of heating sources of the rod, 111-122 [Zbl 07624793]
\textit{Konstantinov, Sergey; Diveev, Askhat}, Evolutionary algorithms for optimal control problem of mobile robots group interaction, 123-136 [Zbl 07624794]
\textit{Aizenberg, Natalia}, Model LSFE with conjectural variations for electricity forward market, 139-153 [Zbl 07624795]
\textit{Kolomeytsev, Yury}, Meta algorithms for portfolio optimization using reinforcement learning, 154-168 [Zbl 07624796]
\textit{Mikhailova, Liudmila}, Simultaneous detection and discrimination of the known number of non-linearly extended alphabet elements in a quasiperiodic sequence, 171-183 [Zbl 07624797]
\textit{Pasechnyuk, Dmitry; Raigorodskii, Andrei M.}, Network utility maximization by updating individual transmission rates, 184-198 [Zbl 07624798]
\textit{Erzin, Adil; Melidi, Georgii; Nazarenko, Stepan; Plotnikov, Roman}, A posteriori analysis of the algorithms for two-bar charts packing problem, 201-216 [Zbl 07624800]
\textit{Koledina, Kamila; Koledin, Sergey; Gubaydullin, Irek}, Pareto frontier in multicriteria optimization of chemical processes based on a kinetic model, 217-229 [Zbl 07624801]
\textit{Malyshev, Dmitry; Rybak, Larisa; Mohan, Santhakumar; Diveev, Askhat; Cherkasov, Vladislav; Pisarenko, Anton}, The method of optimal geometric parameters synthesis of two mechanisms in the rehabilitation system on account of relative position, 230-245 [Zbl 07624802]
\textit{Menshikova, Olga; Sedush, Anna; Polyudova, Daria; Yaminov, Rinat; Menshikov, Ivan}, Laboratory analysis of the social and psychophysiological aspects of the behaviour of participants in the lemons market game, 246-257 [Zbl 07624803]
\textit{Tormagov, Timofey; Rapoport, Lev}, Coverage path planning for 3D terrain with constraints on trajectory curvature based on second-order cone programming, 258-272 [Zbl 07624804]
\textit{Zasukhin, Sergey; Zasukhina, Elena}, Determination of hydrological model parameters by Newton method, 273-285 [Zbl 07624805]QPALM: a proximal augmented Lagrangian method for nonconvex quadratic programshttps://zbmath.org/1496.900322022-11-17T18:59:28.764376Z"Hermans, Ben"https://zbmath.org/authors/?q=ai:hermans.ben"Themelis, Andreas"https://zbmath.org/authors/?q=ai:themelis.andreas"Patrinos, Panagiotis"https://zbmath.org/authors/?q=ai:patrinos.panagiotisSummary: We propose QPALM, a nonconvex quadratic programming (QP) solver based on the proximal augmented Lagrangian method. This method solves a sequence of inner subproblems which can be enforced to be strongly convex and which therefore admit a unique solution. The resulting steps are shown to be equivalent to inexact proximal point iterations on the extended-real-valued cost function, which allows for a fairly simple analysis where convergence to a stationary point at an \(R\)-linear rate is shown. The QPALM algorithm solves the subproblems iteratively using semismooth Newton directions and an exact linesearch. The former can be computed efficiently in most iterations by making use of suitable factorization update routines, while the latter requires the zero of a monotone, one-dimensional, piecewise affine function. QPALM is implemented in open-source C code, with tailored linear algebra routines for the factorization in a self-written package LADEL. The resulting implementation is shown to be extremely robust in numerical simulations, solving all of the Maros-Meszaros problems and finding a stationary point for most of the nonconvex QPs in the Cutest test set. Furthermore, it is shown to be competitive against state-of-the-art convex QP solvers in typical QPs arising from application domains such as portfolio optimization and model predictive control. As such, QPALM strikes a unique balance between solving both easy and hard problems efficiently.Distributionally robust two-stage stochastic programminghttps://zbmath.org/1496.900432022-11-17T18:59:28.764376Z"Duque, Daniel"https://zbmath.org/authors/?q=ai:duque.daniel"Mehrotra, Sanjay"https://zbmath.org/authors/?q=ai:mehrotra.sanjay"Morton, David P."https://zbmath.org/authors/?q=ai:morton.david-pCorrection to: ``Complexity of stochastic dual dynamic programming''https://zbmath.org/1496.900442022-11-17T18:59:28.764376Z"Lan, Guanghui"https://zbmath.org/authors/?q=ai:lan.guanghuiWe point out some corrections needed in the author's paper [ibid. 191, No. 2 (A), 717--754 (2022; Zbl 1489.90082)].Convex optimization for finite-horizon robust covariance control of linear stochastic systemshttps://zbmath.org/1496.900482022-11-17T18:59:28.764376Z"Kotsalis, Georgios"https://zbmath.org/authors/?q=ai:kotsalis.georgios"Lan, Guanghui"https://zbmath.org/authors/?q=ai:lan.guanghui"Nemirovski, Arkadi S."https://zbmath.org/authors/?q=ai:nemirovski.arkadi-sIn this paper the following finite horizon, discrete-time control problem is considered:
\begin{gather*}
x_{0}=z+s_{0},\quad x_{t+1}=A_{t}x_{t}+B_{t}u_{t}+B_{t}^{(d)}d_{t}+B_{t} ^{(s)}e_{t}\\
y_{t}=C_{t}x_{t}+D_{t}^{(d)}d_{t}+D_{t}^{(s)}e_{t},\qquad0\leq t\leq N-1
\end{gather*}
where \(x_{t}\in \mathbb{R}^{n_{x}}\) are states, \(u_{t}\in \mathbb{R}^{n_{u}}\) are controls, \(y_{t}\in \mathbb{R}^{n_{y}}\) are observable outputs, \(z\in \mathbb{R}^{n_{x}}\) and \(d_{t}\in \mathbb{R}^{n_{d}}\) are deterministic factors, \(s_{0}\in \mathbb{R}^{n_{x}}\), \(e_{t}\in \mathbb{R}^{n_{e}}\) are stochastic factors. The deterministic disturbance vector \(\zeta=\left[ z;d_{0};\ldots;d_{N-1}\right] \) is assumed to lie in an ellitop (for example, a finite intersection of centered at the origin ellipsoids and elliptic cylinders). \(A_{t},B_{t},\ldots,D_{t}^{(s)}\) are known matrices.
For this problem a procedure for designing control policies that guarantee some performance specifications is developed. The parameters of the policy are obtained as solutions to an explicit convex program.
Reviewer: Nicolas Hadjisavvas (Ermoupoli)The saddle point problem of polynomialshttps://zbmath.org/1496.900522022-11-17T18:59:28.764376Z"Nie, Jiawang"https://zbmath.org/authors/?q=ai:nie.jiawang"Yang, Zi"https://zbmath.org/authors/?q=ai:yang.zi"Zhou, Guangming"https://zbmath.org/authors/?q=ai:zhou.guangmingSummary: This paper studies the saddle point problem of polynomials. We give an algorithm for computing saddle points. It is based on solving Lasserre's hierarchy of semidefinite relaxations. Under some genericity assumptions on defining polynomials, we show that: (i) if there exists a saddle point, our algorithm can get one by solving a finite hierarchy of Lasserre-type semidefinite relaxations; (ii) if there is no saddle point, our algorithm can detect its nonexistence.Inertial accelerated primal-dual methods for linear equality constrained convex optimization problemshttps://zbmath.org/1496.900562022-11-17T18:59:28.764376Z"He, Xin"https://zbmath.org/authors/?q=ai:he.xin"Hu, Rong"https://zbmath.org/authors/?q=ai:hu.rong"Fang, Ya-Ping"https://zbmath.org/authors/?q=ai:fang.yapingSummary: In this paper, we propose an inertial accelerated primal-dual method for the linear equality constrained convex optimization problem. When the objective function has a ``nonsmooth + smooth'' composite structure, we further propose an inexact inertial primal-dual method by linearizing the smooth individual function and solving the subproblem inexactly. Assuming merely convexity, we prove that the proposed methods enjoy \(\mathcal{O}(1/k^2)\) convergence rate on the objective residual and the feasibility violation in the primal model. Numerical results are reported to demonstrate the validity of the proposed methods.Fast convergence of generalized forward-backward algorithms for structured monotone inclusionshttps://zbmath.org/1496.900582022-11-17T18:59:28.764376Z"Maingé, Paul-Emile"https://zbmath.org/authors/?q=ai:mainge.paul-emileSummary: We develop rapidly convergent forward-backward algorithms for computing zeroes of the sum of finitely many maximally monotone operators. A modification of the classical forward-backward method for two general operators is first considered, by incorporating an inertial term (close to the acceleration techniques introduced by Nesterov), a constant relaxation factor and a correction
term. In a Hilbert space setting, we prove the weak convergence to equilibria of the iterates \((x_n)\), with worst-case rates of \(o(n^{-1})\) in terms of both the discrete velocity and the fixed point residual, instead of the classical rates of \(\mathcal O(n^{-1/2})\) established so far for related algorithms. Our procedure
is then adapted to more general monotone inclusions and a fast primal-dual algorithm is proposed for solving convex-concave saddle point problems.Running primal-dual gradient method for time-varying nonconvex problemshttps://zbmath.org/1496.900682022-11-17T18:59:28.764376Z"Tang, Yujie"https://zbmath.org/authors/?q=ai:tang.yujie"Dall'Anese, Emiliano"https://zbmath.org/authors/?q=ai:dallanese.emiliano"Bernstein, Andrey"https://zbmath.org/authors/?q=ai:bernstein.andrey"Low, Steven"https://zbmath.org/authors/?q=ai:low.steven-hCharacterization results of solutions in interval-valued optimization problems with mixed constraintshttps://zbmath.org/1496.900692022-11-17T18:59:28.764376Z"Treanţă, Savin"https://zbmath.org/authors/?q=ai:treanta.savinSummary: In this paper, we establish some characterization results of solutions associated with a class of interval-valued optimization problems with mixed constraints. More precisely, we investigate the connections between the LU-optimal solutions of the considered interval-valued variational control problem and the saddle-points associated with an interval-valued Lagrange functional corresponding to a modified interval-valued variational control problem. The main derived resuts are accompanied by illustrative examples.Connectedness of solution sets for generalized vector equilibrium problems via free-disposal sets in complete metric spacehttps://zbmath.org/1496.900892022-11-17T18:59:28.764376Z"Shao, Chong-Yang"https://zbmath.org/authors/?q=ai:shao.chongyang"Peng, Zai-Yun"https://zbmath.org/authors/?q=ai:peng.zaiyun"Xiao, Yi-Bin"https://zbmath.org/authors/?q=ai:xiao.yibin"Zhao, Yong"https://zbmath.org/authors/?q=ai:zhao.yongSummary: In this paper, the connectedness of solution sets for generalized vector equilibrium problems via free-disposal sets (GVEPVF) in complete metric space is discussed. Firstly, by virtue of Gerstewitz scalarization functions and oriented distance functions, a new scalarization function \(\omega\) is constructed and some properties of it are given. Secondly, with the help of \(\omega \), the existence of solutions for scalarization problems (GVEPVF)\(_\omega\) and the relationship between the solution sets of (GVEPVF)\(_\omega\) and (GVEPVF) are obtained. Then, under some suitable assumptions, sufficient conditions of (path) connectedness of solution sets for (GVEPVF) are established. Finally, as an application, the connectedness results of \(E\)-efficient solution set for a class of vector programming problems are derived. The obtained results are new, and some examples are given to illustrate the main results.Block coordinate descent for smooth nonconvex constrained minimizationhttps://zbmath.org/1496.900922022-11-17T18:59:28.764376Z"Birgin, E. G."https://zbmath.org/authors/?q=ai:birgin.ernesto-g"Martínez, J. M."https://zbmath.org/authors/?q=ai:martinez.jose-marioSummary: At each iteration of a block coordinate descent method one minimizes an approximation of the objective function with respect to a generally small set of variables subject to constraints in which these variables are involved. The unconstrained case and the case in which the constraints are simple were analyzed in the recent literature. In this paper we address the problem in which block constraints are not simple and, moreover, the case in which they are not defined by global sets of equations and inequations. A general algorithm that minimizes quadratic models with quadratic regularization over blocks of variables is defined and convergence and complexity are proved. In particular, given tolerances \(\delta >0\) and \(\varepsilon >0\) for feasibility/complementarity and optimality, respectively, it is shown that a measure of \((\delta,0)\)-criticality tends to zero; and the number of iterations and functional evaluations required to achieve \((\delta,\varepsilon)\)-criticality is \(O(\varepsilon^{-2})\). Numerical experiments in which the proposed method is used to solve a continuous version of the traveling salesman problem are presented.Fully stable well-posedness and fully stable minimum with respect to an admissible functionhttps://zbmath.org/1496.901002022-11-17T18:59:28.764376Z"Zhu, Jiangxing"https://zbmath.org/authors/?q=ai:zhu.jiangxing"Arthur Köbis, Markus"https://zbmath.org/authors/?q=ai:kobis.markus-arthur"Hu, Chunhai"https://zbmath.org/authors/?q=ai:hu.chunhai"He, Qinghai"https://zbmath.org/authors/?q=ai:he.qinghai"Li, Jiaxiong"https://zbmath.org/authors/?q=ai:li.jiaxiongThe authors consider parametric optimization problems in an infinite-dimensional setting where the parameters are taken from a metric space. In particular, parameter and tilt perturbations are taken into account. Generalizing known concepts such as stable well-posedness and fully stable Hölder minimum, the authors introduce the two new notions of fully stable well-posedness and fully stable minimum and discuss some of their relationships. Moreover, they present conditions for full stability as well as for full minimality.
Reviewer: Jan-Joachim Rückmann (Bergen)Bregman proximal point algorithm revisited: a new inexact version and its inertial varianthttps://zbmath.org/1496.901082022-11-17T18:59:28.764376Z"Yang, Lei"https://zbmath.org/authors/?q=ai:yang.lei.3"Toh, Kim-Chuan"https://zbmath.org/authors/?q=ai:toh.kimchuanMean field games of controls: finite difference approximationshttps://zbmath.org/1496.910112022-11-17T18:59:28.764376Z"Achdou, Yves"https://zbmath.org/authors/?q=ai:achdou.yves"Kobeissi, Ziad"https://zbmath.org/authors/?q=ai:kobeissi.ziadSummary: We consider a class of mean field games in which the agents interact through both their states and controls, and we focus on situations in which a generic agent tries to adjust her speed (control) to an average speed (the average is made in a neighborhood in the state space). In such cases, the monotonicity assumptions that are frequently made in the theory of mean field games do not hold, and uniqueness cannot be expected in general. Such model lead to systems of forward-backward nonlinear nonlocal parabolic equations; the latter are supplemented with various kinds of boundary conditions, in particular Neumann-like boundary conditions stemming from reflection conditions on the underlying controled stochastic processes. The present work deals with numerical approximations of the above mentioned systems. After describing the finite difference scheme, we propose an iterative method for solving the systems of nonlinear equations that arise in the discrete setting; it combines a continuation method, Newton iterations and inner loops of a bigradient like solver. The numerical method is used for simulating two examples. We also make experiments on the behaviour of the iterative algorithm when the parameters of the model vary.Numerical solution of mean field problem with limited management resourcehttps://zbmath.org/1496.910122022-11-17T18:59:28.764376Z"Kornienko, V."https://zbmath.org/authors/?q=ai:kornienko.vladimir-v|kornienko.viktoria-s|kornienko.v-f|kornienko.v-g|kornienko.v-n"Shaydurov, V."https://zbmath.org/authors/?q=ai:shaidurov.vladimir-victorovichSummary: The paper presents a computational algorithm for solving problem formulated in terms of Mean Field Game theory with limited management resource. The Mean Field equilibrium leads to a coupled system of two parabolic partial differential equations: Fokker-Planck-Kolmogorov and Hamilton-Jacobi-Bellman ones with an additional constraint in the form of the inequality. The article is devoted to the discrete approximation of these equations and reformulating the discrete statement in the form of the saddle point problem with the condition of complementary slackness. An iterative algorithm is presented for solving the obtained discrete problem with justification of the convergence of its elements. The convergence of the algorithm as a whole is illustrated by a numerical example.A terrorism-based differential game: Nash differential gamehttps://zbmath.org/1496.910252022-11-17T18:59:28.764376Z"Megahed, Abd El-Monem A."https://zbmath.org/authors/?q=ai:megahed.abd-el-monem-aSummary: In this paper, we investigate the problem of combating terrorism by the government, which is one of the most serious problems that direct governments and countries. We formulate the problem and use the Nash approach of a differential game to obtain the optimal strategies for combating terrorism. We study the relationship between each of the government' strategies and terrorism when the government is on the defensive (reactive), and we study when the government expects terrorist attacks and develops its strategies to combat terrorism. Also, we study the relationship between government activity and its strategies as well as government activity and the strategy of terrorist organizations.A variational formulation of network games with random utility functionshttps://zbmath.org/1496.910322022-11-17T18:59:28.764376Z"Passacantando, Mauro"https://zbmath.org/authors/?q=ai:passacantando.mauro"Raciti, Fabio"https://zbmath.org/authors/?q=ai:raciti.fabioSummary: We consider a class of games played on networks in which the utility functions consist of both deterministic and random terms. In order to find the Nash equilibrium of the game we formulate the problem as a variational inequality in a probabilistic Lebesgue space which is solved numerically to provide approximations for the mean value of the random equilibrium. We also numerically compare the solution thus obtained, with the solution computed by solving the deterministic variational inequality derived by taking the expectation of the pseudo-gradient of the game with respect to the random parameters.
For the entire collection see [Zbl 1485.65002].Optimal control methodology for the counter-terrorism strategies: the relaxation based approachhttps://zbmath.org/1496.910422022-11-17T18:59:28.764376Z"Azhmyakov, Vadim"https://zbmath.org/authors/?q=ai:azhmyakov.vadim"Verriest, Erik I."https://zbmath.org/authors/?q=ai:verriest.erik-i"Bonilla, Moises"https://zbmath.org/authors/?q=ai:bonilla.moises-e"Pickl, Stefan"https://zbmath.org/authors/?q=ai:pickl.stefan-wolfgangSummary: This paper deals with a novel application of the advanced optimal control techniques to a counter-terrorism optimal decision-making problem. The recently developed model of the counter-terrorism security policies (see [\textit{L. Armijo}, Pac. J. Math. 16, 1--3 (1966; Zbl 0202.46105)]) naturally incorporates a competing interest group dynamics. The optimal dynamics can be found in this case by considering a specific optimal control problem. in our work we propose an essential formal improvement of the above applied optimal control problem and its numerical treatment. We formulate a suitable phase-constrained optimal control problem for the initial counter-terrorism model and study a specific relaxation of this model. The proposed \(\beta\)-relaxation approach makes it possible to eliminate some model inconsistencies and to design an optimal counter-terrorism strategy. We develop a new numeric algorithm for the improved dynamic optimization problem. We next study some analytic properties of this algorithm. The resulting computational technique involves a gradient based method in combination with the proposed relaxation scheme (see [\textit{V. Azhmyakov} and \textit{W. Schmidt}, J. Optim. Theory Appl. 130, No. 1, 61--77 (2006; Zbl 1135.49020)]). The obtained numerical approach is finally applied to an illustrative example.Variational inequalities and general equilibrium modelshttps://zbmath.org/1496.910532022-11-17T18:59:28.764376Z"Donato, Maria Bernadette"https://zbmath.org/authors/?q=ai:donato.maria-bernadette"Maugeri, Antonino"https://zbmath.org/authors/?q=ai:maugeri.antonino"Milasi, Monica"https://zbmath.org/authors/?q=ai:milasi.monica"Villanacci, Antonio"https://zbmath.org/authors/?q=ai:villanacci.antonioSummary: We deal with the study of several general equilibrium models by using the variational inequality theory. The theory of variational inequalities was introduced in the sixties of the past century by \textit{G. Fichera} [Atti Accad. Naz. Lincei, Mem., Cl. Sci. Fis. Mat. Nat., VIII. Ser., Sez. I 7, 91--140 (1964; Zbl 0146.21204)], and
\textit{J. L. Lions} and \textit{G. Stampacchia} [Commun. Pure Appl. Math. 20, 493--519 (1967; Zbl 0152.34601)], as an innovative and effective method to solve equilibrium problems arising in mathematical physics. Afterward this theory turned out as a powerful tool, and it was used to analyze different kinds of equilibrium problems.
For the entire collection see [Zbl 1483.00042].Dynamics and control of delayed rumor propagation through social networkshttps://zbmath.org/1496.910692022-11-17T18:59:28.764376Z"Ghosh, Moumita"https://zbmath.org/authors/?q=ai:ghosh.moumita"Das, Samhita"https://zbmath.org/authors/?q=ai:das.samhita"Das, Pritha"https://zbmath.org/authors/?q=ai:das.prithaSummary: Investigation of rumor spread dynamics and its control in social networking sites (SNS) has become important as it may cause some serious negative effects on our society. Here we aim to study the rumor spread mechanism and the influential factors using epidemic like model. We have divided the total population into three groups, namely, ignorant, spreader and aware. We have used delay differential equations to describe the dynamics of rumor spread process and studied the stability of the steady-state solutions using the threshold value of influence which is analogous to the basic reproduction number in disease model. Global stability of rumor prevailing state has been proved by using Lyapunov function. An optimal control system is set up using media awareness campaign to minimize the spreader population and the corresponding cost. Hopf bifurcation analyses with respect to time delay and the transmission rate of rumors are discussed here both analytically and numerically. Moreover, we have derived the stability region of the system corresponding to change of transmission rate and delay values.Group chase and escape. Fusion of pursuits-escapes and collective motionshttps://zbmath.org/1496.920022022-11-17T18:59:28.764376Z"Kamimura, Atsushi"https://zbmath.org/authors/?q=ai:kamimura.atsushi"Ohira, Toru"https://zbmath.org/authors/?q=ai:ohira.toruThe book under review is a result of collaboration of researchers with the experience in two different areas of mathematics. The interest of the first author in the development of theoretical models explaining consistent growth and division of primitive cells and the work of the second author with the models of a random walk with delay led to the publication of the book combining classical mathematical problems of chases and escapes with an emerging research field studying collective motions of ``self-driven'' particles.
In the introductory Chapter 1, the origins of the two fields, chases and escapes and collective motions, are discussed. Both have a long and interesting history. The formulation of the first chase and escape problem is attributed to Leonardo da Vinci who considered a cat-chasing-a-mouse problem and the first studies of multi-particle systems in physics are related to the work of Johannes Kepler who investigated relations between the geometry and motion of six planets based on the astronomical data collected by Tycho Brahe. The authors argue that a model of a ``group chase and escape'' can be viewed either as a ``simple extension of the traditional chase and escape problems to multiple players'' or as an ``extension of the self-propelled particles into a mixture of two groups with different motives''. Some of the recent research developments are briefly described along with potential applications of group chases and escapes and open problems.
Chapter 2 introduces the reader to representative problems including chases and escapes along straight lines and along circular paths in the plane. The extensions to 3D space allowing targets to move along circular cylindrical helices and equiangular spiral helices are also considered.
Fundamental concepts of statistical mechanics (phase transition, order parameters, symmetry, critical exponents, spin models) are discussed in Chapter 3. These are used to explain phase transition in equilibrium models and collective motion in non-equilibrium models. Two theoretical models are analysed, the Vicsek flocking model and the optimal velocity model. The latter one describes traffic jams on highways in a 1D case whereas its 2D extension models collective motion of pedestrians and animals.
Chapter 4 is fundamental for the understanding of a concept of group chase and escape which fuses two research fields introduced in Chapters 2 and 3. The basic model is built on the following simple rule: a chaser tries to get closer to the nearest target and a target attempts to get away from the nearest chaser. The lifetimes of targets are calculated and their dependence on the number of chasers and initial targets are explored. For the quantitative analysis of chasing processes, a simple classification focusing on ``one-particle-to-many-opponents situations'' is considered. In the final part of the chapter, recent developments in group chases and escapes are discussed from the view points of abilities (modifications of the model allowing agents to detect the opponents' positions), reactions (modifications of the model reflecting how the targets that are captured affect their species and mortality), and motions (modifications of the rules restricting spatial motion of particles).
The final Chapter 5 addresses a number of open problems in group chases and escapes and provides possible directions for future developments in the field. The crucial role of boundary conditions is first discussed; the authors argue that the choice of an appropriate kind of boundary condition is highly nontrivial; it depends on the aspects of the system one wants to study and impacts the understanding of the general behavior of group chase and escape. Challenging issues regarding the characterisation and distribution of chasing patterns and their dependence on the initial conditions are also addressed. Two representative examples for flocking and traffic flow are considered to illustrate the development of macroscopic models which describe collective behavior. Potential applications of group chases and escapes considered in this chapter include hunting in nature, optimisation problems, and a sketch of a ``general perspective on the power of living together.'' There are four appendices in the book illustrating interesting variations and extensions of chases and escapes: Discrete search game on graphs, Chase and escape with delays, Virtual stick balancing, and minority games. The rich list of references includes 105 items.
The book is well-written, explanations are concise and transparent, and the excellent quality of the print with many illustrations in color matches the quality of the exposition. It is an interesting introduction to a new area of research bridging pursuits-escapes with collective motions where, according to the authors, ''the successful fusion of the two topics remains to be achieved.''
Reviewer: Svitlana P. Rogovchenko (Kristiansand)Optimal control of an HIV infection model with logistic growth, celluar and homural immune response, cure rate and cell-to-cell spreadhttps://zbmath.org/1496.920402022-11-17T18:59:28.764376Z"Akbari, Najmeh"https://zbmath.org/authors/?q=ai:akbari.najmeh"Asheghi, Rasoul"https://zbmath.org/authors/?q=ai:asheghi.rasoulSummary: In this paper, we propose an optimal control problem for an HIV infection model with cellular and humoral immune responses, logistic growth of uninfected cells, cell-to-cell spread, saturated infection, and cure rate. The model describes the interaction between uninfected cells, infected cells, free viruses, and cellular and humoral immune responses. We use two control functions in our model to show the effectiveness of drug therapy on inhibiting virus production and preventing new infections. We apply Pontryagin maximum principle to study these two control functions. Next, we simulate the role of optimal therapy in the control of the infection by numerical simulations and AMPL software.Modeling and dynamic analysis of novel coronavirus pneumonia (COVID-19) in Chinahttps://zbmath.org/1496.921122022-11-17T18:59:28.764376Z"Guo, Youming"https://zbmath.org/authors/?q=ai:guo.youming"Li, Tingting"https://zbmath.org/authors/?q=ai:li.tingting(no abstract)Mathematical modeling of the spread of COVID-19 among different age groups in Morocco: optimal control approach for intervention strategieshttps://zbmath.org/1496.921162022-11-17T18:59:28.764376Z"Kada, Driss"https://zbmath.org/authors/?q=ai:kada.driss"Kouidere, Abdelfatah"https://zbmath.org/authors/?q=ai:kouidere.abdelfatah"Balatif, Omar"https://zbmath.org/authors/?q=ai:balatif.omar"Rachik, Mostafa"https://zbmath.org/authors/?q=ai:rachik.mostafa"Labriji, El Houssine"https://zbmath.org/authors/?q=ai:labriji.elhoussineSummary: In this article, we study the transmission of COVID-19 in the human population, notably between potential people and infected people of all age groups. Our objective is to reduce the number of infected people, in addition to increasing the number of individuals who recovered from the virus and are protected. We propose a mathematical model with control strategies using two variables of controls that represent respectively, the treatment of patients infected with COVID-19 by subjecting them to quarantine within hospitals and special places and using masks to cover the sensitive body parts. Pontryagin's Maximum principle is used to characterize the optimal controls and the optimality system is solved by an iterative method. Finally, numerical simulations are presented with controls and without controls. Our results indicate that the implementation of the strategy that combines all the control variables adopted by the World Health Organization (WHO), produces excellent results similar to those achieved on the ground in Morocco.Decentralized control for multi-sensors networked systems with different transmission delays and packet dropoutshttps://zbmath.org/1496.930142022-11-17T18:59:28.764376Z"Zhang, Qianqian"https://zbmath.org/authors/?q=ai:zhang.qianqian"Qi, Qingyuan"https://zbmath.org/authors/?q=ai:qi.qingyuan"Tan, Cheng"https://zbmath.org/authors/?q=ai:tan.cheng"Wong, Wing Shing"https://zbmath.org/authors/?q=ai:wong.wing-shingSummary: In this paper, the linear quadratic (LQ) optimal decentralized control and stabilization problems are investigated for multi-sensors networked control systems (MSNCSs) with multiple controllers of different information structure. Specifically, for a MSNCS, in view of the packet dropouts and the transmission delays, each controller may access different information sets. To begin with, the sufficient and necessary solvability conditions for the LQ decentralized control problems are developed. Consequently, for the purpose of deriving the optimal decentralized control strategy, an innovative orthogonal decomposition method is proposed to decouple the forward and backward stochastic difference equations (FBSDEs) from the maximum principle. In the following, we show that the optimal decentralized controller can be calculated according to a set of Riccati-type equations. Finally, a stabilizing controller is derived for the stabilization problem.Riemannian optimization model order reduction method for general linear port-Hamiltonian systemshttps://zbmath.org/1496.930262022-11-17T18:59:28.764376Z"Li, Zi-Xue"https://zbmath.org/authors/?q=ai:li.zi-xue"Jiang, Yao-Lin"https://zbmath.org/authors/?q=ai:jiang.yaolin"Xu, Kang-Li"https://zbmath.org/authors/?q=ai:xu.kangliSummary: This paper presents a Riemannian optimal model order reduction method for general linear stable port-Hamiltonian systems based on the Riemannian trust-region method. We consider the \(\mathcal{H}_2\) optimal model order reduction problem of the general linear port-Hamiltonian systems. The problem is formulated as an optimization problem on the product manifold, which is composed of the set of skew symmetric matrices, the manifold of the positive definite matrices, the manifold of the positive semidefinite matrices with fixed rank and the Euclidean space. To solve the optimal problem, the Riemannian geometry of the product manifold is given. Moreover, the Riemannian gradient and the Riemannian Hessian of the objective function are derived. Furthermore, we propose the Riemannian trust-region method for the optimization problem and introduce the truncated conjugate gradient method to solve the trust-region subproblem. Finally, the numerical experiments illustrate the efficiency of the proposed method.Optimizing oblique projections for nonlinear systems using trajectorieshttps://zbmath.org/1496.930272022-11-17T18:59:28.764376Z"Otto, Samuel E."https://zbmath.org/authors/?q=ai:otto.samuel-e"Padovan, Alberto"https://zbmath.org/authors/?q=ai:padovan.alberto"Rowley, Clarence W."https://zbmath.org/authors/?q=ai:rowley.clarence-wAnalytical synthesis of an amplitude-constrained controllerhttps://zbmath.org/1496.930292022-11-17T18:59:28.764376Z"Ashchepkov, L. T."https://zbmath.org/authors/?q=ai:ashchepkov.leonid-timofeevichSummary: We consider a time-invariant optimal control problem of a new linear-quadratic type on the positive time axis with an amplitude constraint on the control. Using sufficient optimality conditions, we find an optimal positional control with a discontinuity on a subspace of the state space. The motion of the closed-loop system on the subspace in the sliding mode is investigated. The exponential stability of the closed-loop system is shown. Examples are given.Stabilization of a class of switched dynamic systems: the Riccati-equation-based approachhttps://zbmath.org/1496.930932022-11-17T18:59:28.764376Z"Bonilla, M."https://zbmath.org/authors/?q=ai:bonilla.moises-e"Aguillón, N. A."https://zbmath.org/authors/?q=ai:aguillon.n-a"Ortiz Castillo, M. A."https://zbmath.org/authors/?q=ai:ortiz-castillo.m-a"Jacques Loiseau, Jean"https://zbmath.org/authors/?q=ai:loiseau.jean-jacques"Malabre, M."https://zbmath.org/authors/?q=ai:malabre.michel"Azhmyakov, V."https://zbmath.org/authors/?q=ai:azhmyakov.vadim"Salazar, S."https://zbmath.org/authors/?q=ai:salazar.sergio|salazar.santiago|salazar.saul-j-cSummary: Our paper deals with the stabilization of a class of time-dependent linear autonomous complex systems with a switched structure. The initially given switched dynamic system is assumed to be controlled by a specific state feedback strategy associated with the linear quadratic regulator (LQR) type control. The proposed control design guarantees stabilization of the closed-loop system for all of the possible location transitions. In the solution procedure of the algebraic Riccati equation related to the LQR control strategy, only the knowledge of the algebraic structure related to the switched system are needed. We prove that the proposed optimal LQR type state feedback control design stabilizes the closed-loop switched system for every possible active location. The theoretical approach proposed in this paper is finally applied to a model of the \textit{single wing quadrotor aircraft}, when changing from its \textit{quadrotor flight envelope} to its \textit{airplane flight envelope}.Robust sub-optimality of linear-saturated control via quadratic zero-sum differential gameshttps://zbmath.org/1496.930962022-11-17T18:59:28.764376Z"Bauso, Dario"https://zbmath.org/authors/?q=ai:bauso.dario"Maggistro, Rosario"https://zbmath.org/authors/?q=ai:maggistro.rosario"Pesenti, Raffaele"https://zbmath.org/authors/?q=ai:pesenti.raffaeleSummary: In this paper, we determine the approximation ratio of a linear-saturated control policy of a typical robust-stabilization problem. We consider a system, whose state integrates the discrepancy between the unknown but bounded disturbance and control. The control aims at keeping the state within a target set, whereas the disturbance aims at pushing the state outside of the target set by opposing the control action. The literature often solves this kind of problems via a linear-saturated control policy. We show how this policy is an approximation for the optimal control policy by reframing the problem in the context of quadratic zero-sum differential games. We prove that the considered approximation ratio is asymptotically bounded by 2, and it is upper bounded by 2 in the case of 1-dimensional system. In this last case, we also discuss how the approximation ratio may apparently change, when the system's demand is subject to uncertainty. In conclusion, we compare the approximation ratio of the linear-saturated policy with the one of a family of control policies which generalize the bang-bang one.Stochastic optimal control over unreliable communication linkshttps://zbmath.org/1496.931252022-11-17T18:59:28.764376Z"Bengtsson, Fredrik"https://zbmath.org/authors/?q=ai:bengtsson.fredrik"Wik, Torsten"https://zbmath.org/authors/?q=ai:wik.torstenSummary: In this paper LQG control over unreliable communication links is derived. That is to say, the communication channels between the controller and the actuators and between the sensors and the controller are unreliable. This is of growing importance as networked control systems and use of wireless communication in control are becoming increasingly common. The problem of how to optimize LQG control in this case is examined in the situation where communication between the components is done with acknowledgments. Previous solutions to finite horizon discrete time hold-input LQG control for this case do not fully utilize the available information. Here a new solution is presented which resolves this limitation. The solution is linear and covers communication channels subject to both packet losses and delays. The new control scheme is compared with previous solutions for LQG control in simulations, which demonstrates that a significant improvement in the cost can be achieved by fully utilizing the available information.On the synthesis of an optimal control for a system of nonlinear differential equations that depend on a semi-Markov processhttps://zbmath.org/1496.931262022-11-17T18:59:28.764376Z"Dil'muradov, N."https://zbmath.org/authors/?q=ai:dilmuradov.n(no abstract)Necessary optimality conditions for singular controls in stochastic Goursat-Darboux systemshttps://zbmath.org/1496.931282022-11-17T18:59:28.764376Z"Mansimov, K. B."https://zbmath.org/authors/?q=ai:mansimov.kamil-bayramali-ogly|mansimov.kamil-bairamali-oglu"Mastaliev, R. O."https://zbmath.org/authors/?q=ai:mastaliev.rashad-ogtai-ogly|mastaliev.rashad-ogtaiovichSummary: We consider an optimal control problem for a stochastic system whose dynamics is described by a second-order hyperbolic stochastic partial differential equation with Goursat boundary conditions. A stochastic analog of Pontryagin's maximum principle is obtained, and singularities in the sense of the control maximum principle are analyzed for optimality.