变分不等式的惯性次梯度外梯度算法

杨志　夏福全

摘要：在实Hilbert空间中提出求解单调变分不等式的惯性次梯度外梯度算法，其中变分不等式的可行集是一个光滑凸函数的水平集.新算法应用惯性加速技巧，迭代过程中对映射F赋值一次，并只需向两个半空间作投影两次.在适当的假设下，证明该算法的弱收敛性.新算法改进和推广相关文献中的相应结果.

关键词：次梯度外梯度算法; 单调; Lipschitz连续; 惯性方法; 变分不等式

中图分类号：O117; O178 文献标志码：A 文章编号：1001-8395（2023）05-0591-10

1预备知识

2算法

3收敛性分析

4数值结果

参考文献

[1] FACCHINEI F， PANG J S. Finite-Dimensional Variational Inequalities and Complementarity Problems[M]. New York：Springer-Verlag，2003.

[2] NAGURNEY A. Network Economic：A Variational Inequality Approach[M]. Berlin：Springer-Verlag，1999.

[3] KARAMARDIAN S. Generalized complementarity problem[J]. Journal of Optimization Theory and Applications，1971，8（3）：161-168.

[4] DUVAUT G， LIONS J L， JOHN C W， et al. Inequalities in Mechanics and Physics[M]. Berlin：Springer-Verlag，1976.

[5] KORPELEVICH G M. An extragradient method for finding saddle points and for other problems[J]. Matecon，1976，12：747-756.

[6] 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.

[7] HE S， WU T. A modified subgradient extragradient method for solving monotone variational inequalities[J]. Journal of Inequalities and Applications，2017，2017（1）：1-14.

[8] CAO Y， GUO K. On the convergence of inertial two-subgradient extragradient method for variational inequality problems[J]. Optimization，2020，69（6）：1237-1253.

[9] 陳家欣，叶明露. 一种新的求解变分不等式的惯性双次梯度外梯度算法[J]. 数学进展，2022，51（1）：165-182.

[10] MALITSKY Y V， SEMENOV V V. An extragradient algorithm for monotone variational inequalities[J]. Cybern Sys Anal，2014，50（2）：271-277.

[11] GIBALI A， VAN HIEU D. A new inertial double-projection method for solving variational inequalities[J]. Journal of Fixed Point Theory and Applications，2019，21（4）：1-21.

[12] BAUSCHKE H H， COMBETTES P L. Convex Analysis and Monotone Operator Theory in Hilbert Spaces[M]. New York：Springer-Verlag，2017.

[13] ZDZISAW O. Weak convergence of the sequence of successive approximations for nonexpansive mappings[J]. Bulletin of the American Mathematical Society，1967，73（4）：591-598.

[14] YE M L. An improved projection method for solving generalized variational inequality problems[J]. Optimization，2018，67（9）：1523-1533.

[15] HE S， XU H K. Uniqueness of supporting hyperplanes and an alternative to solutions of variational inequalities[J]. Journal of Global Optimization，2013，57（4）：1375-1384.

An  Inertial Subgradient Extragradient Algorithm for Solving

Variational InequalitiesYANG Zhi，XIA Fuquan（School of Mathematical Sciences， Sichuan Normal University， Chengdu 610066， Sichuan）

Abstract：In this paper， we propose a new inertial subgradient extragradient algorithm for solving monotone variational inequalities in Hilbert space， where the feasible set of variational inequality is the level set of a smooth convex function. The new algorithm uses the inertial acceleration technique. The value of F is calculated once during per iteration， only needs to project to two half spaces twice. Under the appropriate assumptions， the weak convergence of the algorithm is proved. The new algorithm improves and generalizes the corresponding results in the relevant literature.

Keywords：subgradient extragradient algorithm; monotone; Lipschitz continuous; inertial method; variational inequalities

2020 MSC：65K15; 90C25; 90C33

（编辑陶志宁）