A PROXIMAL POINT-TYPE ALGORITHM FOR PSEUDOMONOTONE EQUILIBRIUM PROBLEMS

A globally convergent algorithm for solving equilibrium prob-lems is proposed. The algorithm is based on a proximal point algorithm(shortly (PPA)) with a positive definite matrix M which is not necessar-ily symmetric. The proximal function in existing (PPA) usually is thegradient of a quadratic function, namely, ▽(kxk 2M ). This leads to a prox-imal point-type algorithm. We first solve pseudomonotone equilibriumproblems without Lipschitzian assumption and prove the convergence ofalgorithms. Next, we couple this technique with the Banach contractionmethod for multivalued variational inequalities. Finally some computa-tional results are given. 1. IntroductionLet C be a nonempty closed convex subset in a real Euclidean space R n andf : C × C → R be a bifunction such that f(x,x) = 0 for every x ∈ C. Weconsider the following equilibrium problemshortly EP(f,C): Find x ∗ ∈ Csuch thatf(x ∗ ,y) ≥ 0 ∀y ∈ C.For the main results of this paper, we assume that:(A.1) f(·,·) is pseudomonotone on C,(A.2) f(x,·) is convex on C for all x ∈ C,(A.3) f is continuous on C ×C,(A.4) The solution set S of EP(f,C) is nonempty,(A.5) The matrix M is positive definite.Equilibrium problems appear frequently in many practical problems arising,for instance, physics, engineering, game theory, transportation, economics andnetwork (see [10, 13]). They become an attractive field for many researchersboth in theory and applications (see [2, 3, 5, 9, 15, 16, 17]). These problemsare models whose formulation includes optimization, variational inequalities,(vector) optimization problems, fixed point problems, saddle point problems