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

Abstract

In this chapter we consider a two-level variational problem: a variational inequality problem over the set of solutions of the equilibrium problem. An example of such a problem is the search for the normal Nash equilibrium. To solve this problem, we propose a new iterative proximal algorithm that combines the ideas of a two-stage proximal point method, adaptability, and scheme of iterative regularization. In contrast to the previously applied rules for choosing the step size, the proposed algorithm does not calculate any values of the bifunction at additional points; it does not require knowledge of information about the Lipschitz constants of the bifunction, the Lipschitz constant and strong monotonicity constant of the operator. For monotone Lipschitzian bifunctions and strongly monotone Lipschitz continuous operators, a theorem on the strong convergence of the algorithm is proved. It is shown that the proposed algorithm is applicable to monotone two-level variational inequalities in real Hilbert spaces.