Modified descent methods for solving the monotone variational inequality problem

Recently, Fukushima proposed a differential optimization framework for solving strictly monotone and continuously differentiable variational inequalities. The main result of this paper is to show that Fukushima’s results can be extended to monotone (not necessarily strictly monotone) and Lipschitz continuous (not necessarily continuously differentiable) variational inequalities, if one is willing to modify slightly the basic algorithmic scheme. The modification applies also to a general descent scheme introduced by Zhu and Marcotte.