A Decomposition Method for Variational Inequalities of the Second Kind with Strongly Inverse-Monotone Operators

Abstract

The present paper deals with the convergence analysis of an iterative method for solving variational inequalities of the second kind with strongly inverse-monotone [1] potential coercive operators and convex nondifferentiable functionals in Hilbert spaces. Problems with operators of monotone type arise in mathematical modeling of a wide class of nonlinear processes (e.g., see [2–7]). There is an extensive literature on approximate methods for such problems. In particular, variational inequalities with strongly monotone operators were considered in [8] as well as in [9], where an iterative method similar to that in [8] was suggested and its convergence was analyzed with the use of a different technique. The present paper is a continuation of [10], where an iterative decomposition method was suggested for solving the variational inequalities in question. This algorithm is based on a reduction of the original variational inequality to the saddle point problem for a modified Lagrange functional with the subsequent use of an Uzawa type algorithm for finding the saddle point. Unlike the decomposition methods suggested in [8, 11], this method does not require the inversion of the original operator. The convergence analysis of this iterative process in [10] was based on a representation of the process in the form of a successive approximation method for finding a fixed point of the nonexpansive operator serving as the transition operator of the process. It was first proved that the fixed point set of this operator coincides with the set of saddle points of the modified Lagrange functional. Then it was shown that the iterative sequence is bounded and its Cesaro means weakly converge to a saddle point of the Lagrange functional. In the present paper, we prove the weak convergence of the iterative sequence to a fixed point of the transition operator. To make this possible, we obtain an estimate stronger that the inequality characterizing the nonexpansive property. Moreover, we prove some additional convergence properties of the iterative sequence. For a strongly monotone Lipschitz continuous operator, we obtain the strong convergence of the iterative method.