A subgradient-type method for the equilibrium problem over the fixed point set and its applications

Abstract

In this article, we consider an equilibrium problem: find a point u∈C such that f(u, y) ≥ 0 for all y∈C, where a continuous function satisfies f(x, x) = 0 for all and is a closed convex set. The existing computational methods for this problem require repetitive use of the metric projection onto C, which is often hard to compute. To relax the computational difficulty caused by the metric projection, we present a way to use any firmly nonexpansive mapping T satisfying in place of the metric projection onto C. The proposed method can be applied soundly to the Nash equilibrium problem in noncooperative games.

Keywords:

• equilibrium problem
• variational inequality problem
• saddle point problem
• Nash equilibrium problem
• firmly nonexpansive mapping
• metric projection
• subgradient-type method

Mathematics Subject Classifications 2000:

• Primary 47H06
• 47J20
• 47J25