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

Authors: Iiduka, Hideaki; Yamada, Isao

Source: Optimization, Volume 58, Number 2, February 2009 , pp. 251-261(11)

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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 [image omitted] satisfies f(x, x) = 0 for all [image omitted] and [image omitted] 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 [image omitted] 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

Document Type: Research article

DOI: http://dx.doi.org/10.1080/02331930701762829

Affiliations: 1: Department of Communications and Integrated Systems, Tokyo Institute of Technology, Tokyo, Japan

Publication date: 2009-02-01