Journal of Guangxi Normal University(Natural Science Edition) ›› 2024, Vol. 42 ›› Issue (1): 128-138.doi: 10.16088/j.issn.1001-6600.2023032404

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Strong Convergence Theorem for Pseudomonotonic Variational Inequality Solution Sets and Quasi-non-expansionary Mapping Fixed Point Sets Common Elements

WANG Yongjie, GAO Xinghui*, FANG Mengkai   

  1. School of Mathematics and Computer Science, Yan’an University, Yan’an Shaanxi 716000, China
  • Received:2023-03-24 Revised:2023-06-02 Online:2024-01-25 Published:2024-01-19

Abstract: A new Tseng outer gradient algorithm for solving pseudomonotonic variational inequality problems is proposed in Hilbert space. Under appropriate conditions, it is proved that the iterative sequence generated by the algorithm strongly converges to the common element of the solution set of pseudomonotonic variational inequality problem and the set of fixed points of the quasi-non-expansion map, and numerical experiments are given to illustrate the effectiveness of the proposed algorithm.

Key words: Hilbert space, variational inequality, pseudo-monotony, subgradient external gradient algorithm, strong convergence

CLC Number:  O177.91
[1] AUBIN J P, EKELAND I. Applied nonlinear analysis[M]. New York: Wiley, 1984.
[2] ZHOU H Y, QIN X L. Fixed points of nonlinear operators[M]. Berlin: De Gruyter, 2020.
[3] KONNOV I. Combined relaxation methods for variational inequalities[M]. Berlin: Springer, 2001.
[4] XU H K. Iterative algorithms for nonlinear operators[J]. Journal of the London Mathematical Society, 2002, 66(1): 240-256.
[5] TAN B, FAN J J, QIN X L. Inertial extragradient algorithms with non-monotonic step sizes for solving variational inequalities and fixed point problems[J]. Advances in Operator Theory, 2021, 6(4): 61.
[6] YEKINI S, OLANIYI S I. Strong convergence result for monotone variational inequalities[J]. Numerical Algorithms, 2017, 76: 259-282.
[7] NOOR M A, On iterative methods for solving a system of mixed variational inequalities[J]. Applicable Analysis, 2008, 87(1): 99-108.
[8] SIBONY M. Méthodes itératives pour les équations et inéquations aux dérivees partielles on linéaires de type monotone[J]. Calcolo, 1970, 7(1): 65-183.
[9] KORPELEVICH G M. The extragradient method for finding saddle points and other problems[J]. Matecon, 1976, 12(4): 747-756.
[10] CENSOR Y, GIBALI A, REICH S. The subgradient extragradient method for solving variational inequalities in Hilbert space[J]. Journal of Optimization Theory and applications, 2011, 148(2): 318-335.
[11] TSENG P. A modified forward-backward splitting method for maximal monotone mappings[J]. SIAM Journal on Control and Optimization, 2000, 38(2): 431-446.
[12] FAN J J, QIN X L. Weak and strong convergence of inertial Tseng’s extragradient algorithms for solving variational inequality problems[J]. Optimization, 2021, 70(5/6): 1195-1216.
[13] THONG D V, HIEU D V. Modified Tseng’s extragradient algorithms for variational inequality problems[J]. Journal of Fixed Point Theory and Applications, 2018, 20(4): 152.
[14] 胡绍涛,王元恒,蔡钢.Hilbert空间上关于伪单调变分不等式问题的新Tseng外梯度算法[J/OL].数学学报(中文版):1-10[2023-03-16]. http://kns.cnki.net/kcms/detail/11.2038.O1.20220613.0916.004.html.
[15] THONG D V, VUONG P T. Modified Tseng’s extragradient methods for solving pseudo-monotone variational inequalities[J]. Optimization, 2019, 68(11): 2202-2226.
[16] 谢忠兵,蔡钢,李肖肖,等.Hilbert空间中求解伪单调变分不等式的自适应次梯度外梯度法[J].数学学报(中文版),2023,66(4): 693-706.
[17] 刘丽平,彭建文.求解变分不等式和不动点问题的公共元的修正次梯度外梯度算法[J].数学物理学报,2022,42(5):15-17-1536.
[18] 杨静,龙宪军.关于伪单调变分不等式与不动点问题的新投影算法[J].数学物理学报,2022,42(3):904-919.
[19] 郭丹妮,蔡钢.变分不等式和不动点问题的新迭代算法[J].数学学报(中文版),2022,65(1):77-88.
[20] 蔡钢.Hilbert空间中关于变分不等式问题和不动点问题的粘性隐式中点算法[J].数学物理学报,2020, 40(2):395-407.
[21] ZHAO T Y, WANG D Q, CENG L C, et al. Quasi-inertial Tseng’s extragradient algorithms for pseudomonotone variational inequalities and fixed point problems of quasi-nonexpansive operators[J]. Numerical Functional Analysis and Optimization, 2021, 42(1): 69-90.
[22] DIZA J B, METCALF F T. On the set of subsequential limit points of successive approximations[J]. Transactions of the American Mathematical Society, 1969, 135: 459-485.
[23] THONG D V, HIEU D V. Inertial subgradient extragradient algorithms with line-search process for solving variational inequality problems and fixed point problems[J]. Numerical Algorithms, 2019, 80(4): 1283-1307.
[24] GOEBEL K, REICH S. Uniform convexity,hyperbolic geometry,and nonexpansive mappings[M]. New York: Marcel Dekker, 1983.
[25] MAINGÉ P E. Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization[J]. Set-Valued Analysis, 2008, 16(7): 899-912.
[26] YANG J, LIU H W. Strong convergence result for solving monotone variational inequalities in Hilbert space[J]. Numerical Algorithms, 2019, 80(3): 741-752.
[27] BLUM E, OETTLI W. From optimization and variational inequalities to equilibrium problems[J]. Mathematics Student, 1994, 63(1/2/3/4): 123-145.
[28] KINDERLEHRER D, STAMPACCHIA G. An introduction to variational inequalities and their applications[M]. Philadelphia: Society for Industrial and Apptied Mathematics, 1980.
[29] LIU L Y, QIN X L. Strong convergence theorems for solving pseudo-monotone variational inequality problems and applications[J]. Optimization, 2022, 71(12): 3603-3626.
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