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

摘要

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 f : R{sup}N × R{sup}N →R satisfies f(x, x)=0 for all x∈R{sup}N and C {is contained in} R{sup}N 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 Fix(T) := {x∈R{sup}N : Tx = x} = C in place of the metric projection onto C. The proposed method can be applied soundly to the Nash equilibrium problem in noncooperative games.