Adaptive Algorithm for Variational Inequality Problem Over the Set of Solutions the Equilibrium Problem

Abstract

In this paper we consider a two-level problem: variational inequality on the set of solutions to the equilibrium problem. Examples of this problem are two-level variational inequality problem and the search of a normal Nash equilibrium. For solving this problem, we propose iterative algorithm which combines ideas of two-stage proximal method, adaptation, and iterative regularization. We obtained the theorem about strong convergence of algorithm for monotone bifunctions of Lipschitz type and strictly monotone Lipschitz operators.