Recent zbMATH articles in MSC 65K15 https://zbmath.org/atom/cc/65K15 2021-06-15T18:09:00+00:00 Werkzeug A modified modulus-based multigrid method for linear complementarity problems arising from free boundary problems. https://zbmath.org/1460.65155 2021-06-15T18:09:00+00:00 "Zhang, Li-Li" https://zbmath.org/authors/?q=ai:zhang.lili "Ren, Zhi-Ru" https://zbmath.org/authors/?q=ai:ren.zhi-ru Summary: The linear complementarity problem arising from a free boundary problem can be equivalently reformulated as a fixed-point equation. We present a modified modulus-based multigrid method to solve this fixed-point equation. This modified method is a full approximation scheme using the modulus-based splitting iteration method as the smoother and avoids the transformation between the auxiliary and the original functions which was necessary in the existing modulus-based multigrid method. We predict its asymptotic convergence factor by applying local Fourier analysis to the corresponding two-grid case. Numerical results show that the W-cycle possesses an \(h\)-independent convergence rate and a linear elapsed CPU time, and the convergence rate of the V-cycle can be improved by increasing the smoothing steps. Compared with the existing modulus-based multigrid method, the modified method is more straightforward and is a standard full approximation scheme, which makes it more convenient and efficient in practical applications. Strong convergence of projected reflected gradient methods for variational inequalities. https://zbmath.org/1460.65083 2021-06-15T18:09:00+00:00 "Maingé, Paul-Emile" https://zbmath.org/authors/?q=ai:mainge.paul-emile Summary: The purpose of this work is to revisit the numerical approach to classical variational inequality problems, with monotone and Lipschitz continuous mapping, by means of a regularized dynamical method. A~main feature of the method is that it formally requires only one projection step onto the feasible set and only one evaluation of the involved mapping (at each iteration), combined with some viscosity-like regularization process. A~strong convergence theorem is established in a general setting that allows the use of varying step-sizes without any requirement of additional projections. We also point out that the considered method in absence of regularization does not generate a Fejér-monotone monotone sequence. So a new analysis is developed for this purpose. Explicit iterative algorithms for solving equilibrium problems on Hadamard manifolds. https://zbmath.org/1460.65075 2021-06-15T18:09:00+00:00 "Ansari, Qamrul Hasan" https://zbmath.org/authors/?q=ai:ansari.qamrul-hasan "Islam, Monirul" https://zbmath.org/authors/?q=ai:islam.monirul Summary: The present paper studies two explicit extragradient-like algorithms for solving equilibrium problems involving a pseudomonotone and Lipschitz-type bifunction in the setting of Hadamard manifolds. A new step size rule has been studied which does not depend on the information of the Lipschitz-type constants. The convergenc and the R-linear rate of convergence of the proposed algorithms are studied. Partial regularization and descent method for a extended primal-dual system. https://zbmath.org/1460.90192 2021-06-15T18:09:00+00:00 "Kim, Jong Kyu" https://zbmath.org/authors/?q=ai:kim.jongkyu|kim.jong-kyu "Salahuddin" https://zbmath.org/authors/?q=ai:salahuddin.k-m|salahuddin.salahuddin|salahuddin.|salahuddin.anjum-r|salahuddin.t|salahuddin.2|salahuddin.m|salahuddin.1 Summary: In this works, we consider a system of variational inequality, which can be regarded as an extension of a primal-dual variational inequality system and involves multivalued mappings. The system does not possess monotonicity properties and the feasible set is unbounded in general. To solve the problem, we propose a completely implementable iterative scheme, whose convergence is proved under certain coercivity type conditions. An exponential integrator-based discontinuous Galerkin method for linear complementarity systems. https://zbmath.org/1460.65086 2021-06-15T18:09:00+00:00 "Wang, Zhengyu" https://zbmath.org/authors/?q=ai:wang.zhengyu "Chen, Xiaojun" https://zbmath.org/authors/?q=ai:chen.xiaojun.1 Summary: The linear complementarity system (LCS) is defined by a linear ordinary differential equation coupled with a finite-dimensional linear complementarity problem (LCP), which has many applications in engineering and economics. In this article, we reformulate the LCS with the boundary condition as an LCP in the Hilbert space of square-integrable functions, and propose a new numerical method for the LCS by using exponential Euler integrator and discontinuous Galerkin approximation. The precision of the proposed method is better than that of the existing time-stepping method in different magnitude of scale. Convergence analysis and numerical experiments are performed to support the arguments. Proximal-like subgradient methods for solving multi-valued variational inequalities. https://zbmath.org/1460.65082 2021-06-15T18:09:00+00:00 "Anh, P. N." https://zbmath.org/authors/?q=ai:pham-ngoc-anh. "Thach, H. T. C." https://zbmath.org/authors/?q=ai:thach.h-t-c "Kim, Jong Kyu" https://zbmath.org/authors/?q=ai:kim.jong-kyu Summary: In this paper, we propose and analyze the convergence of a new algorithm for solving monotone and Lipschitz continuous multi-valued variational inequalities by using proximal operator. By choosing suitable parameters of proximal steps and of subgradient stepsizes, we show that the convergence of the algorithm does not require the prior knowledge of Lipschitz continuous constant of cost operator. A new method for solving variational inequalities and fixed points problems of demi-contractive mappings in Hilbert spaces. https://zbmath.org/1460.49005 2021-06-15T18:09:00+00:00 "Chen, Xue" https://zbmath.org/authors/?q=ai:chen.xue "Wang, Zhong-bao" https://zbmath.org/authors/?q=ai:wang.zhongbao "Chen, Zhang-you" https://zbmath.org/authors/?q=ai:chen.zhangyou The paper presents an algorithm for finding a common element of the fixed points set of a demi-contraction and of the set of solutions of a variational inequality with pseudomonotone operator in a Hilbert space. The convergence and the efficiency of the algorithm are discussed. Some illustrative numerical experiments are provided. Reviewer: Petru Jebelean (Timişoara) Mann-type algorithms for variational inequality problems and fixed point problems. https://zbmath.org/1460.65085 2021-06-15T18:09:00+00:00 "Viet Thong, Duong" https://zbmath.org/authors/?q=ai:duong-viet-thong. "Van Hieu, Dang" https://zbmath.org/authors/?q=ai:dang-van-hieu. Summary: The purpose of this paper is to investigate the problem of finding a common element of the solution set of a variational inequality problem for a monotone, Lipschitz-continuous mapping and set of fixed point of a quasi-nonexpansive mapping. We introduce Mann-type Tseng's extragradient-like approximation method which is based on so-called inertial Tseng's extragradient method and Mann method. We establish two weak convergence theorems for two iterative sequences generated by these methods. Our results extend and improve some related results in the literatures. Stochastic recursive inclusions in two timescales with nonadditive iterate-dependent Markov noise. https://zbmath.org/1460.62136 2021-06-15T18:09:00+00:00 "Yaji, Vinayaka G." https://zbmath.org/authors/?q=ai:yaji.vinayaka-g "Bhatnagar, Shalabh" https://zbmath.org/authors/?q=ai:bhatnagar.shalabh Summary: In this paper, we study the asymptotic behavior of a stochastic approximation scheme on two timescales with set-valued drift functions and in the presence of nonadditive iterate-dependent Markov noise. We show that the recursion on each timescale tracks the flow of a differential inclusion obtained by averaging the set-valued drift function in the recursion with respect to a set of measures accounting for both averaging with respect to the stationary distributions of the Markov noise terms and the interdependence between the two recursions on different timescales. The framework studied in this paper builds on a recent work by Ramaswamy and Bhatnagar, by allowing for the presence of nonadditive iterate-dependent Markov noise. As an application, we consider the problem of computing the optimum in a constrained convex optimization problem, where the objective function and the constraints are averaged with respect to the stationary distribution of an underlying Markov chain. Further, the proposed scheme neither requires the differentiability of the objective function nor the knowledge of the averaging measure. Modified subgradient extragradient method for a family of pseudomonotone equilibrium problems in real a Hilbert space. https://zbmath.org/1460.65084 2021-06-15T18:09:00+00:00 "Rehman, Habib Ur" https://zbmath.org/authors/?q=ai:rehman.habib-ur "Pakkaranang, Nuttapol" https://zbmath.org/authors/?q=ai:pakkaranang.nuttapol "Kuman, Poom" https://zbmath.org/authors/?q=ai:kuman.poom "Cho, Yeol Je" https://zbmath.org/authors/?q=ai:cho.yeol-je Summary: In this paper, we proposed a modified subgradient extragradient method for dealing with pseudomonotone equilibrium problems involving Lipschitz-type condition on a cost bifunction in a real Hilbert space. The weak convergence theorem for the method is precisely provided based on the standard assumptions on the cost bifunction. For a numerical experiment, we consider the well-known Nash-Cournot oligopolistic equilibrium models and other examples to support our established convergence results. Subgradient algorithms for Ky Fan inequalities and fixed point problems. https://zbmath.org/1460.65081 2021-06-15T18:09:00+00:00 "Wang, Ke" https://zbmath.org/authors/?q=ai:wang.ke.4|wang.ke|wang.ke.2|wang.ke.3|wang.ke.1 "Yao, Yonghong" https://zbmath.org/authors/?q=ai:yao.yonghong "Liou, Yeong-Cheng" https://zbmath.org/authors/?q=ai:liou.yeongcheng Summary: In this paper, we present a subgradient iterative algorithm for finding a common element of the solution of the Ky Fan inequalities and the fixed point of a pseudocontractive operator in Hilbert spaces. Strong convergence of the suggested algorithm is given under some suitable assumptions.