Predictor-corrector Methods For Solving Equilibrium Problems

Equilibrium problems which was by Blum-Oettli in 1993 is a kind of nonsmooth optimiza-tion problem. Classical studies focus on dual problems. existence of solutions and relationship to other problems etc, few papers have studied efficient algorithms to solve equilibrium problems. So we need a method not only to solve equilibrium problems, but also be operated effectively taking into account of practical-problems.This paper presents a prediction-correct method for solving nonsmooth convex equilibrium problems based on the auxiliary problem principle. In the algorithm each iteration operate two steps. One step serves to predict the next point; the other helps to correct the new prediction. At the same time, we present convergence analysis under the case of perfect foresight and imperfect foresight. Moreover, we introduce a stopping criterion which gives rise to V-stationary points. Finally, we employ the algorithm for solving variational inequalities.This paper is an extension over the related work of [2] and [4], the main contributions can be summarized as follows. First of all, we extend the coefficient of approximate function fromμ∈(0,1) to∈(?)\{0}; Secondly, approximate function need not to satisfy the conditions (C1)-(C3) in [4]; Thirdlly, it is much easier to calaulate the optimal solution in this paper than in [2] about the step of prediction. Finally, this paper present a stop criteria.