Research On Convex Optimization Problems And Related Problems

In this thesis,we study two related problems:convex optimization problems and equilibrium problems.This thesis is divided into three chapters. It is organized as follows:In chapter 1,we introduce some marks and basic definitions.In chapter 2,we consider the equivalence of a constraint,qualification,conjugate formulae for a sum of two convex functions,andε-subdifferential calculus. Firstly.we study the equivalence of the sum of two convex functions in locally convex spaces.One of the convex functions is a composite of a convex function with a.continuous linear mapping.This equivalence is extended to the case where the continuous linear mapping is a continuous convex mapping.In chapter 3,we consider equilibrium problems.We provide a system of vector version of Ekeland's theorem related to the system of vector equilibrium problems.Via the system of vector Ekeland's variational principle,existence results for the system of vector equilibrium problems are proved in both compact and noncompact domains,without the assumption of any convexity of the domains and the functions involved.As an application,we deduce the solutions of systems of vector variational inequalities.