Study On P-K Convergence And Related Properties Of Solutions For Set Optimization Problems

In this thesis,we devote to studying the stability of solutions for set optimization problems,including the following three aspects:Painleve-Kuratowski convergence of approximate solution sets for set optimization problems via general ordering sets,Berge lower semicontinuity of parametric generalized vector quasi-equilibrium problems under improvement set mappings and connectedness of solution sets for generalized vector equilibrium problems via free-disposal sets.The paper is divided into six chapters as follows:In Chapter 1,we briefly introduce the related situation of set optimization problems,and mainly introduce Painleve-Kuratowski convergence of solution sets for set optimization problems,continuity of solution mapping for vector equilibrium problems and connectedness of solution sets,which are three research directions related to this paper.In Chapter 2,we mainly give some basic concepts and related theories which are used in the research of this paper.It mainly includes continuity,convexity of mapping and two kinds of nonlinear scalarization functions.In Chapter 3,we mainly consider the connectedness of the solution sets for generalized vector equilibrium problems via free-disposal sets.Firstly,based on two classes of classical nonlinear scalarization functions(the Gerstewitz functions and the oriented distance functions),we introduce a new nonlinear scalarization function?,and discuss basic properties of ?.Then,by means of the new function ?,we obtain scalarization problems of the original problem,and discuss the existence of solution and the closedness and convexity of solution set for scalarization problems.Next,we establish connectedness and path connectedness results of solution sets for generalized vector equilibrium problems via free-disposal sets.Finally,as an application of the main conclusions,we derive the connectedness results of E-efficient solution sets for nonlinear programming problems via free-disposal sets.In Chapter 4,we mainly study the Berge lower semicontinuity of solution mapping for parametric generalized vector quasi-equilibrium problems under improvement set mappings.Firstly,inspired by the cone-valued mapping C(·),we give the concept of improvement set mapping and propose the parametric generalized vector quasi-equilibrium problems under improvement set mappings.Secondly,based on the improved mapping,we introduces a nonlinear scalarization function ?,and discuss some properties of ?.With the help of function ?,we obtain scalarization problems of the original problem.Moreover,we study the relationship of solution sets between the original problem and scalarization problems.Finally,based on a key assumption and the results obtained above,we establish sufficient conditions for the Berge lower semicontinuity of solution mapping for parametric generalized vector quasi-equilibrium problems under improvement set mappings.In Chapter 5,we focus on the Painleve-Kuratowski convergence of approximate solution sets for set optimization problems via general ordering sets.Firstly,by virtue of the set order relation defined by a general ordered set,we give a class of set optimization problems,namely set optimization problems via general ordering sets.Meanwhile,we give some concepts of minimal solution,approximate minimal solution,weak minimal solution and approximate weak minimal solution,respectively.Then,we discuss the relationships between the four kinds of solution sets and the related properties of the level set of objective mapping.Finally,with the help of related properties of recession cone,we obtain the sufficient conditions of upper Painleve-Kuratowski convergence of approximate minimal solution sets,upper Painleve-Kuratowski convergence and lower Painleve-Kuratowski convergence of approximate weak minimal solution sets.In Chapter 6,we summarize the results of this paper and make some outlook.