變分不等式的慣性次梯度外梯度算法

楊志　夏福全

摘要：在實(shí)Hilbert空間中提出求解單調(diào)變分不等式的慣性次梯度外梯度算法，其中變分不等式的可行集是一個(gè)光滑凸函數(shù)的水平集.新算法應(yīng)用慣性加速技巧，迭代過程中對映射F賦值一次，并只需向兩個(gè)半空間作投影兩次.在適當(dāng)?shù)募僭O(shè)下，證明該算法的弱收斂性.新算法改進(jìn)和推廣相關(guān)文獻(xiàn)中的相應(yīng)結(jié)果.

關(guān)鍵詞：次梯度外梯度算法; 單調(diào); Lipschitz連續(xù); 慣性方法; 變分不等式

中圖分類號：O117; O178 文獻(xiàn)標(biāo)志碼：A 文章編號：1001-8395（2023）05-0591-10

1預(yù)備知識

2算法

3收斂性分析

4數(shù)值結(jié)果

參考文獻(xiàn)

[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，葉明露. 一種新的求解變分不等式的慣性雙次梯度外梯度算法[J]. 數(shù)學(xué)進(jìn)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

（編輯陶志寧）