Proximal Point Approach and Approximation of Variational Inequalities

Abstract

A general approach to analyze convergence of the proximal-like methods for variational inequalities with set-valued maximal monotone operators is developed. It is oriented to methods coupling successive approximation of the variational inequality with the proximal point algorithm as well as to related methods using regularization on a subspace and weak regularization. This approach also covers so-called multistep regularization methods, in which the number of proximal iterations in the approximated problems is controlled by a criterion characterizing these iterations as to be effective. The conditions on convergence require control of the exactness of the approximation only in a certain region of the original space. Conditions ensuring linear convergence of the methods are established.