Optimality Conditions For Several Approximate Solutions To Set-valued Equilibrium Problems In Locally Convex Spaces

Vector equilibrium problem covers many typical mathematical problems,such as vector optimization,variational inequality,vector Nash equilibrium,vector complementarity and so on.Due to the universality and unity of the problems involved and the profundity of solving them,vector equilibrium has become a hot issue in the field of operational research and non-linear analysis.This thesis consists of four chapters investigating several approximate solutions to set-valued equilibrium problems in Hausdorff locally convex topological linear spaces.The relationships between these approximate solutions and their corresponding effective solutions were discussed,and their necessary and sufficient optimality conditions were established.Chapter 1 is an introduction,which presents the background of approximate solutions to set-valued equilibrium problems,their current research situation and some relative basic concepts.In chapter 2,the concepts.of Benson efficient solutions and approximate Benson efficient solutions to set-valued equilibrium problems were introduced,and the relationships between them were discussed.Under the assumption of the near cone-subconvexlike,the Kuhn-Tucker-type and Lagrange-type optimality conditions of them were established respectively by using the separation theorem for convex sets.The efficient solutions of approximate Henig,approximate weakly and approximate globally to unconstrained set-valued equilibrium problems were researched in chapter 3.The relationships between these approximate solutions and their corresponding efficient solutions were obtained.The optimality conditions of them,without any convexity assumption,were established by the virtue of a nonlinear functional.In chapter 4,the constrained set-valued equilibrium problems'efficient solutions of approximate Henig,approximate weakly and approximate globally were investigated by applying a nonlinear functional.Based on this,the optimality conditions of them were established.Another sufficient optimality condition of approximate Henig efficient solutions was established especially by weakening the functional properties.