Existence Of Solutions And Proximal Point Algorithm For Generalized Mixed Equilibrium Problem

As a generalization of variational inequality problems, equilibrium problems have been wildly and deeply used in many fields, such as economic equilibrium, physics, nonlinear programming, modern control. Generalized mixed equilibrium problems are important extension from equilibrium problems, which contact closely with control theory, game theory, optimization and some nonlinear analysis problems in engineering, and have extensive application background. In this article we will study from two aspects, which are the existence for generalized mixed implicit vector equilibrium problems and hybrid proximal methods for generalized mixed equilibrium problems.The main research results of this article are summarized as follows.Chapter 2 The existence of solutions for a class of generalized mixed implicit vector equilibrium problem will be discussed. First of all, in the mapping without any monotonicity conditions, by using the KKM theorem, we prove the existence theorem of solutions for generalized mixed implicit vector equilibrium problem; Then, only in set convexity instead of compactness requirement circumstances, by making use of the maximal element lemma, we prove the existence theorem of solutions for generalized mixed implicit vector equilibrium problem, and the obtained results extend and improve some corresponding results in the literature.Chapter 3 We mainly consider a generalized mixed equilibrium problem by making use of hybrid proximal point methods for finding a common solution of some optimization-related problems. First of all, combing the extragradient method and approximate proximal point method, we construct an algorithm for solving a generalized mixed equilibrium problem and a generalized variational inequality problem at the same time. Furthermore, for finding a common solution of two different generalized mixed equilibrium problems, we put forward another algorithm based on an approximate proximal point method. Finally, under suitable conditions, we prove the sequences generated by the above algorithms are global convergent.