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

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。

A remark on the convergence of the two-subgradient extragradient algorithm for the variational inequality problem

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.