求解变分不等式与不动点问题公共解的新Tseng型外梯度算法

摘要/Abstract

摘要：

该文在Hilbert空间中提出一种新Tseng型外梯度算法, 用以求解一致连续伪单调映射的变分不等式问题与具有半封闭性拟非扩张映射的不动点问题的公共解. 在一定的假设条件下, 证明了算法所生成的序列的强收敛性. 文章最后对算法进行数值实验, 验证了算法的有效性.

关键词: 变分不等式, 不动点问题, 伪单调映射, 拟非扩张映射, 强收敛

Abstract:

In this paper, we introduce a new inertial Tseng-like extragradient algorithm for finding a common element of the set of solutions of a variational inequalitiy problem with a pseudomonotone and non-Lipschitz continuous mapping and the set of a fixed point problem with a quasi-nonexpansive mapping in Hilbert spaces. The strong convergence of the sequences generated by the algorithm is proved under some suitable assumptions imposed on the parameters. Finally, numercial experiments are carried out on the algorithm to verify the effectiveness of the algorithm.

Key words: Variational inequalitiy, Fixed point problem, Pseudomonotone mapping, Quasi-nonexpansive mapping, Strong convergence

中图分类号:

O244

段洁, 夏福全. 求解变分不等式与不动点问题公共解的新Tseng型外梯度算法[J]. 数学物理学报, 2023, 43(1): 274-290.

Duan Jie, Xia Fuquan. A New Tseng-like Extragradient Algorithm for Common Solutions of Variational Inequalities and Fixed Point Problems[J]. Acta mathematica scientia,Series A, 2023, 43(1): 274-290.

图/表 6

表1

表2

表3

表4

表5

表6

参考文献 23

[1]	Goldstein A A. Convex programming in Hilbert space. Bull New Ser Am Math Soc, 1964, 70(5): 109-112
[2]	Korpelevich G M. The extragradient method for finding saddle points and other problems. Ekonomikai Matematicheskie Metody, 1976, 12: 747-756
[3]	Tseng P. A modified forward-backward splitting method for maximal monotone mapping. SIAM J Control Optim, 2000, 38(2): 431-446 doi: 10.1137/S0363012998338806
[4]	Cai G, Yekini S, Iyiola O S. Inertial Tseng's extragradient method for solving variational inequality problems of pseudo-monotone and non-lipschitz operators. J Ind Manag Optim, 2021, 1: 1-30 doi: 10.3934/jimo.2005.1.1
[5]	Moudafi A. Viscosity approximation methods for fixed-points problems. J Math Anal Appl, 2000, 241: 46-55 doi: 10.1006/jmaa.1999.6615
[6]	Yamada I. The hybrid steepest descent method for the variational inequality problem over the intersection of fixed point sets of nonexpansive mappings//Butnariu D, Censor Y, Reich S, Eds. Inherently Parallel Algorithms in Feasibility and Optimization and Their Application. New York: Elservier, 2001: 473-504
[7]	Halpern B. Fixed points of nonexpanding maps. Bull Amer Math, 1967, 73: 957-961
[8]	Ishikawa S. Fixed points by a new iteration method. Proc Amer Math Soc, 1974, 44: 147-150 doi: 10.1090/S0002-9939-1974-0336469-5
[9]	Marino G, Xu H K. A general iterative method for nonexpansive mappings in Hilbert spaces. J Math Anal Appl, 2006, 318: 43-52 doi: 10.1016/j.jmaa.2005.05.028
[10]	Tian M. A general iterative method for nonexpansive mappings in Hilbert spaces. Nonlinear Anal, 2010, 73(3): 689-694 doi: 10.1016/j.na.2010.03.058
[11]	Ke Y F, Ma C F. The generalized viscosity implicit rules of nonexpansive mappings in Hilbert spaces. Fixed Point Theory Appl, 2015, 2015: 190 doi: 10.1186/s13663-015-0439-6
[12]	Nadezhkina N, Takahashi W. Weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings. J Optim Theory Appl, 2006, 128(1): 191-201 doi: 10.1007/s10957-005-7564-z
[13]	Thong D V, Hieu D V. Inertial subgradient extragradient algorithms with line-search process for solving variational inequality problems and fixed point problems. Numer Algor, 2019, 80(4): 1283-1307 doi: 10.1007/s11075-018-0527-x
[14]	Ceng L C, Shang M J. Hybrid inertial subgradient extragradient methods for variational inequalities and fixed point problems involving asymptotically nonexpansive mappings. Optimization, 2021, 70(4): 715-740 doi: 10.1080/02331934.2019.1647203
[15]	Goebel K, Reich S. Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings. New York: Marcel Dekker, 1984
[16]	Cottle R W, Yao J C. Pseudo-monotone complementarity problems in Hilbert space. J Optim Theory Appl, 1992, 72(2): 281-295
[17]	Xu H K. Iterative algorithms for nonlinear operators. J London Math Soc, 2002, 66: 240-256 doi: 10.1112/S0024610702003332
[18]	Xu H K, Kim T H. Convergence of hybrid steepest-descent methods for variational inequalities. J Optim Theory Appl, 2003, 119(1): 185-201 doi: 10.1023/B:JOTA.0000005048.79379.b6
[19]	Glowinski R, Lions J L, Trémolières R. Numerical Analysis of Variational Inequalities. Amsterdam: Elsevier, 1981
[20]	Iusem A, Otero R G. Inexact versions of proximal point and augmented lagrangian algorithms in Banach spaces. Numer Funct Anal Optim, 2001, 22: 609-640 doi: 10.1081/NFA-100105310
[21]	Denisov S V, Semenov V V, Chabak L M. Convergence of the modified extragradient method for variational inequalities with non-lipschitz operators. Cybern Syst Anal, 2015, 51: 757-765 doi: 10.1007/s10559-015-9768-z
[22]	He B S, Liao L Z. Improvements of some projection methods for monotone nonlinear variational inequalities. J Optim Theory Appl, 2002, 112(1): 111-128 doi: 10.1023/A:1013096613105
[23]	Sun D F. A projection and contraction method for the nonlinear complementarity problem and it's extensions. Math Numer Sinica, 1994, 16(3): 183-194