Banach空间上变分不等式的一个超梯度方法

An Extragradient-Type Method for Variational Inequalities on Banach Spaces

把王宜举等人[Modified extragradient—type method for variational inequali—ties and verification of the existence of solutions,J.Optim.Theory Appl.,2003,119:167-183]在欧氏空间上求解变分不等式的一个超梯度型方法推广到Banach空间.变分不等式中的算子不要求是一致连续的,其主要优点在于不管变分不等式是否有解,算法都是可执行的.此外,变分不等式的可解性可以通过算法产生的序列的性态来刻画.在适当的条件下,算法产生的序列强收敛于变分不等式的一个解,这是Bregman距离意义下离初始点最近的解.本文的主要结果推广和改善了近来文献中的相应结果.

In this paper, an extragradient-type method proposed by Wang, Xiu and Zhang [Modified extragradient-type method for variational inequalities and verification of the existence of solutions, J. Optim. Theory Appl., 2003, 119： 167-183] for solving variational inequalities in Euclidean spaces is extended to Banach spaces. The operator involved in the variational inequality is not necessarily uniformly continuous. The main advantage lies in that the proposed algorithm is well defined no matter whether the variational inequality problem has a solution or not. Furthermore, the existence of the solution to the variational inequality problem can be verified through the behavior of the generated sequence. Under some suitable assumptions, the sequence generated by the proposed method is strongly convergent to the solution of the variational inequality, which is closest to the initial iterate in the sense of Bregman distance. The main results presented in this paper generalize and improve the recent ones in the literature.