Convergence of Extragradient Algorithm with Monotone Step Size Strategy for Variational Inequalities and Operator Equations

Abstract

Variational inequalities and operator equations in an infinite dimensional Hilbert space with additional conditions in the terms of inclusion in the set of fixed points of a given operator are considered. For approximate solution of the problems, a new iterative algorithm that is a superposition of a modified Korpelevich extragradient algorithm with monotone step size strategy, which does not require knowledge of the Lipschitz operator constant, and the Krasnoselsky–Mann scheme for the approximation of fixed points, is proposed. In contrast to the previously used rules for choosing the step size, the proposed algorithm does not perform additional calculations for the operator values and the projections mapping. The algorithm was investigated using the theory of iterative processes of the Fejer type. The weak convergence of the algorithm for problems with pseudomonotone, Lipschitz continuous and sequentially weakly continuous operators and quasi nonexpansive operators, which specify additional conditions, is proved. Previously, similar results on weak convergence were known only for variational inequalities with monotone, Lipschitz continuous operators and with nonexpansive operators, which specify additional conditions.