运筹学学报 >
2021 , Vol. 25 >Issue 2: 144 - 148
DOI: https://doi.org/10.15960/j.cnki.issn.1007-6093.2021.02.012
关于求解变分不等式问题的2-次梯度外梯度算法收敛性的一个补注
收稿日期: 2019-11-19
网络出版日期: 2021-05-06
基金资助
山东省自然科学基金(ZR2018MA019);国家自然科学基金(11271226)
A remark on the convergence of the two-subgradient extragradient algorithm for the variational inequality problem
Received date: 2019-11-19
Online published: 2021-05-06
Yair Censor，Aviv Gibali和Simeon Reich为求解变分不等式问题提出了2-次梯度外梯度算法。关于此算法的收敛性，作者给出了部分证明，有一个问题：由算法产生的迭代点列能否收敛到变分不等式问题的一个解上，没有得到解决。此问题作为一个公开问题在文章“Extensions of Korpelevich's extragradient method for the variational inequalityproblem in Euclidean space”(Optimization，61(9)：1119-1132，2012)中被提出。在这篇简短的补注性文章中，对所提出的问题给出了答案：由算法产生的迭代点列能收敛到变分不等式问题的一个解上。给出2-次梯度外梯度算法的全局收敛性的一个完整证明，证明了从任意起始点开始，由算法产生的迭代点列都能收敛到变分不等式问题的一个解上。
关键词： 变分不等式问题; 2-次梯度外梯度算法; 收敛性
屈彪, 徐伟, 王新艳 . 关于求解变分不等式问题的2-次梯度外梯度算法收敛性的一个补注[J]. 运筹学学报, 2021 , 25(2) : 144 -148 . DOI: 10.15960/j.cnki.issn.1007-6093.2021.02.012
The two-subgradient extragradient algorithm was proposed by Yair Censor, Aviv Gibali and Simeon Reich for solving the variational inequality problem. A question about the convergence of this algorithm, that is, whether the sequences generated by the algorithm converge to a solution of the variational inequality problem, was raised as an open problem in the paper "Extensions of Korpelevich's extragradient method for the variational inequality problem in Euclidean space" (Optimization, 61(9): 1119-1132, 2012). Our goal in this short remark is to give an answer to this question and give an integrated proof of the full convergence of the algorithm.
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