Research On The Stability And Connectedness Of Solution Sets In Vector Optimization And Related Problems

Vector optimization is one of the main research fields of optimization theory and applications.The research of this problem involves many branches,including convex analysis,nonlinear analysis and variational analysis.Simultaneously,the vector optimization theory has been widely used in many fields,such as financial management,engineering planning,transportation,ecological protection and decision theory.Therefore,there has theoretical and practical significance in the research of vector optimization.This paper focuses on the stability and connectivity of the solution set of vector optimization and its related problems.The full text is divided into seven chapters:In the first chapter,the research background and significance of vector optimization and its related problems are summarized,and the research status of continuity and connectedness of solution sets in this kind of problem is briefly introduced.Meanwhile,the main research content of this paper is introduced.In the second chapter,the basic symbols,definitions,properties,propositions and lemmas are provided.In the third chapter,a kind of level set mapping of parametric set-valued mapping is introduced,and its semicontinuity by-Hausdorff continuity instead of continuity in sense of Berge are obtained.Then,with the help of the stability and appropriate conditions of the level set mapping,the semicontinuity of the minimal solution set mapping for parameter set-valued vector optimization are discussed.In the fourth chapter,by means of the nonlinear scalarization function,the union relationship between the weak efficient solution set of vector optimization problem and a series of solutions of scalar optimization problem is established.Then,by some properties of the solution set of the scalarization problem and the union relationship between the solution sets,the connectedness and path connectedness of the weak efficient solution set of the vector optimization problem are obtained.In the fifth chapter,a nonlinear regular weak separation function is established by the property of the nonlinear scalarization function,and the nonlinear regular weak separation function is proved to be the gap function of the cone constrained vector varaitonal inequality.Then,by the continuity of the gap function,the continuity and Hausdorff continuity of solution set of the parametric cone constrained vector variational inequality are obtained.In the sixth chapter,two definitions of new monotonicity are given by analyzing a kind of partial order relation and its nonlinear scalarization function.By means of the monotonic conditions,the continuity of the solution mapping of the parametric set optimization problem is discussed.In the seventh chapter,the main research works of this paper are summarized,and some further issues worth thinking about are put forward.