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extragradient algorithms for solving variational inequalities on Hadamard manifolds

NUMERICAL algorithmS 2025年 1-34页

作者： Tan, Bing Chen, Jiawei Li, Songxiao Ou, Xiaoqing Shantou Univ Dept Math Shantou 515063 Peoples R China Southwest Univ Sch Math & Stat Chongqing 400715 Peoples R China Sichuan Prov Univ Key Lab Bridge Nondestruct Detecting & Engn Comp Zigong Peoples R China North Minzu Univ Sch Math & Informat Sci Yinchuan 750021 Peoples R China

We provide five techniques for solving variational inequality problems on Hadamard manifolds that are based on the adaptive extragradient method. These algorithms operate adaptively, eliminating the need for prior knowledge of the Lipschitz constant associated with the vector field. Furthermore, the iterative sequences produced by the algorithms are shown to converge to the solution of the problem under the conditions that the vector fields are pseudo-monotone and Lipschitz continuous. Additionally, we establish global error bounds and R-linear convergence rates when the vector fields exhibit strong pseudo-monotonicity. Lastly, the theoretical results are illustrated with two numerical instances.

关键词： Variational inequality extragradient algorithm Adaptive stepsize Hadamard manifolds Pseudo-monotone vector field

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Two modified extragradient algorithms for solving variational inequalities

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JOURNAL OF GLOBAL OPTIMIZATION 2020年第1期78卷 91-106页

作者： Hai, Trinh Ngoc Hanoi Univ Sci & Technol Sch Appl Math & Informat 1 Dai Co Viet Hanoi Vietnam

In this paper, we discuss two modified extragradient methods for variational inequalities. The first one can be applied when the Lipschitz constant of the involving operator is unknown. In contrast to the work by Hieu and Thong (J Glob Optim 70:385-399, 2018) and by Khanh (Numer Funct Anal Optim 37:1131-1143, 2016), the new algorithm does not require its step-sizes tending to zero. This feature helps to speed up our method. The second algorithm solves variational inequalities with non-Lipschitz continuous operators. Under the pseudomonotonicity assumption, the proposed algorithm converges to a solution of the problem. In contrast to other solution methods for this class of problems, the new algorithm does not require the step sizes being square summable. Some numerical experiments show that the new algorithms are more effective than the existing ones.

关键词： Variational inequality Lipschitz continuity Pseudomonotonicity extragradient algorithm

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Modified extragradient algorithms for solving monotone variational inequalities and fixed point problems

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OPTIMIZATION 2020年第9期69卷 1987-2002页

作者： Zhao, Xiaopeng Yao, Yonghong Tiangong Univ Sch Math Sci Tianjin Peoples R China North Minzu Univ Key Lab Intelligent Informat & Big Data Proc Ning Yinchuan Peoples R China

In this paper, we investigate the monotone variational inequalities and fixed point problems in Hilbert spaces. Two modified extragradient algorithms are presented for finding a common element of the set of fixed points of a pseudocontractive operator and the set of solutions of the variational inequality problem. Weak and strong convergence of the suggested algorithms are proved.

关键词： extragradient algorithm fixed point variational inequality pseudocontraction

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Inertial extragradient algorithms for Solving Equilibrium Problems

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ACTA MATHEMATICA VIETNAMICA 2019年第3期44卷 639-663页

作者： Nguyen The Vinh Le Dung Muu Univ Transport & Commun Dept Math 3 Cau Giay St Hanoi Vietnam Thang Long Univ TIMAS Hanoi Vietnam VAST Inst Math Hanoi Vietnam

We introduce two new algorithms based upon the extragradient and inertial methods for solving pseudomonotone equilibrium problems in real Hilbert spaces. Strong converge and weak convergence of the proposed algorithms are established under some mild assumptions. Numerical results show that the proposed algorithms are more efficient than some existing methods for equilibrium problems.

关键词： Inertial method extragradient algorithm Variational inequality Equilibrium problem Pseudomonotone Lipschitz-type inequality

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Strong convergence of modified inertial extragradient methods for non-Lipschitz continuous variational inequalities and fixed point problems

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COMPUTATIONAL & APPLIED MATHEMATICS 2024年第1期43卷 1-25页

作者： Zhang, Huan Liu, Xiaolan Deng, Jia Sun, Yan Sichuan Univ Sci & Engn Coll Math & Stat Zigong 643000 Sichuan Peoples R China South Sichuan Ctr Appl Math Zigong 643000 Sichuan Peoples R China

In this paper, we present two modified algorithms for finding a common element of the solution set of a quasimonotone variational inequality and the fixed point set of demicontractive mapping. The main advantage of such iterative methods is that it does not require Lipschitz continuity and the mapping only needs to be quasimonotonicity. Therefore, our algorithms can be applied to more general variational inequalities. Several numerical experiments are provided to verify the preponderance and efficiency of the proposed algorithms.

关键词： extragradient algorithm Quasimonotone variational inequality Fixed point set Demicontractive mapping

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extragradient and linesearch methods for solving split feasibility problems in Hilbert spaces

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MATHEMATICAL METHODS IN THE APPLIED SCIENCES 2019年第12期42卷 4343-4359页

作者： Jouymandi, Zeynab Moradlou, Fridoun Sahand Univ Technol Dept Math Tabriz Iran

Utilizing the Tikhonov regularization method and extragradient and linesearch methods, some new extragradient and linesearch algorithms have been introduced in the framework of Hilbert spaces. In the presented algorithms, the convexity of optimization subproblems is assumed, which is weaker than the strong convexity assumption that is usually supposed in the literature, and also, the auxiliary equilibrium problem is not used. Some strong convergence theorems for the sequences generated by these algorithms have been proven. It has been shown that the limit point of the generated sequences is a common element of the solution set of an equilibrium problem and the solution set of a split feasibility problem in Hilbert spaces. To illustrate the usability of our results, some numerical examples are given. Optimization subproblems in these examples have been solved by FMINCON toolbox in MATLAB.

关键词： equilibrium problem extragradient algorithm linesearch algorithm Lipschitz-type condition split feasibility problem

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A Tseng extragradient method for solving variational inequality problems in Banach spaces

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NUMERICAL algorithmS 2022年第2期89卷 769-789页

作者： Oyewole, O. K. Abass, H. A. Mebawondu, A. A. Aremu, K. O. Univ KwaZulu Natal Sch Math Stat & Comp Sci Durban South Africa DSI NRF Ctr Excellence Math & Stat Sci CoE MaSS Johannesburg South Africa Usmanu Danfodiyo Univ Sokoto Dept Math PMB 2346 Sokoto Sokoto State Nigeria Sefako Magkatho Hlth Sci Univ Dept Math & Appl Math POB 60 ZA-0204 Ga Rankuwa South Africa

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.

关键词： Variational inequality Pseudomonotone operator Strong convergence Banach space extragradient algorithm Step-size rule

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The extragradient method for quasi-monotone variational inequalities

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OPTIMIZATION 2022年第9期71卷 2519-2528页

作者： Salahuddin Jazan Univ Dept Math Jazan Saudi Arabia

The goal of this paper is to study quasi-monotone variational inequality problems in infinite dimensional Hilbert spaces and to prove that the iterative sequence generated by an extragradient algorithm converges weakl... 详细信息

关键词： Variational inequality problems extragradient algorithm quasi-monotonicity weak convergence Hilbert spaces

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Computational Errors of the extragradient Method for Equilibrium Problems

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BULLETIN OF THE MALAYSIAN MATHEMATICAL SCIENCES SOCIETY 2019年第5期42卷 2835-2858页

作者： Pham Ngoc Anh Nguyen Duc Hien Pham Minh Tuan Posts & Telecommun Inst Technol Dept Sci Fundamentals Hanoi Vietnam Duy Tan Univ Off Sci Res & Technol Da Nang Vietnam Acad Mil Sci & Technol Hanoi Vietnam

Our aim in this paper is to study variants and computational errors of the extragradient method for solving equilibrium problems. First, we consider convergence of the method when domains in the auxiliary subproblems of the extragradient algorithm are replaced by outer and inner approximation polyhedra. Then, computational errors are showed under the asymptotic optimality condition, but the bifunction must satisfy certain Lipschitz-type continuous conditions. Next, by using Armijo-type linesearch techniques commonly used in variational inequalities, we obtain an approximation linesearch algorithm without Lipschitz continuity. Convergence analysis of the algorithms is considered under mild conditions on the iterative parameters.

关键词： Equilibrium problems Semicontinuous extragradient algorithm Computational errors

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Ultra-Low-Complexity algorithms With Structurally Optimal Multi-Group Multicast Beamforming in Large-Scale Systems

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IEEE TRANSACTIONS ON SIGNAL PROCESSING 2023年 71卷 1626-1641页

作者： Zhang, Chong Dong, Min Liang, Ben Univ Toronto Dept Elect & Comp Engn Toronto ON M5S 1A1 Canada Ontario Tech Univ Dept Elect Comp & Software Engn Oshawa ON Canada

In this work, we propose ultra-low-complexity design solutions for multi-group multicast beamforming in large-scale systems. For the quality-of-service (QoS) problem, by utilizing the optimal multicast beamforming structure obtained recently in [2], we convert the original problem into a non-convex weight optimization problem of a lower dimension and propose two fast first-order algorithms to solve it. Both algorithms are based on successive convex approximation (SCA) and provide fast iterative updates to solve each SCA subproblem. The first algorithm uses a saddle point reformulation in the dual domain and applies the extragradient method with an adaptive step-size procedure to find the saddle point with simple closed-form updates. The second algorithm adopts the alternating direction method of multipliers (ADMM) method by converting each SCA subproblem into a favorable ADMM structure. The structure leads to simple closed-form ADMM updates, where the problem in each update block can be further decomposed into parallel subproblems of small sizes, for which closed-form solutions are obtained. We also propose efficient initialization methods to obtain favorable initial points that facilitate fast convergence. Furthermore, taking advantage of the proposed fast algorithms, for the max-min fair (MMF) problem, we propose a simple closed-form scaling scheme that directly uses the solution obtained from the QoS problem, avoiding the conventional computationally expensive method that iteratively solves the inverse QoS problem. We further develop lower and upper bounds on the performance of this scaling scheme. Simulation results show that the proposed algorithms offer near-optimal performance with substantially lower computational complexity than the state-of-the-art algorithms for large-scale systems.

关键词： Array signal processing Multicast algorithms Signal processing algorithms Quality of service Interference Optimization Computational complexity Alternating direction method of multipliers extragradient algorithm large-scale optimization low complexity multicast beamforming optimal beamforming structure

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