Research On The Well-posedness And The Semicontinuity Of Solution Set Mapping Of Set Optimization Problems

In this thesis,pointwise Levitin-Polyak well-posedness and generalized l-well-posedness with perturbation of set optimization problems are discussed.On one hand,the pointwise Levitin-Polyak well-posedness of set optimization problems are considered.Firstly,we introduce the notions of pointwise Levitin-Polyak L-well-posedness,pointwise Levitin-Polyak Bwell-posedness and pointwise Levitin-Polyak DH-well-posedness of set optimization problems.Then the relationships among these three kinds of pointwise Levitin-Polyak well-posedness are established.We also derive the relationships between these three kinds of pointwise LevitinPolyak well-posedness of set optimization problems and the corresponding Levitin-Polyak well-posedness of scalar optimization problems by a nonlinear scalarization function.Particularly,the equivalence between pointwise Levitin-Polyak DH-well-posedness of set optimization problems and Levitin-Polyak well-posedness of scalar optimization problems is obtained under some mild conditions.On the other hand,the generalized l-well-posedness with perturbation of set optimization problems is considered.Firstly,we introduce the notions of generalized l-well-posedness with perturbation and l-well-setness with perturbation of set optimization problems.Then the relationships between generalized l-well-posedness with perturbation and l-well-setness with perturbation of set optimization problems are obtained.And we study the relationships between semicontinuity of l-minimal solution mappings of parametric set optimization problems and the generalized l-well-posedness with perturbation of set optimization problems.Furthermore,the equivalence between generalized l-well-posedness with perturbation of set optimization problem and generalized well-posedness with perturbation of a scalar optimization problem is derived.The thesis is divided into five chapters and organized as follows:In chapter 1,the development and present researches on the related topics of optimization problems are firstly recalled.Then the motivations and the main research work of this thesis are also given.In chapter 2,some notations,definitions,and basis assumptions and properties are recalled.The chapter 3 includes three parts.The first part is the preliminary knowledge required for this chapter.The relationships among three kinds of pointwise Levitin-Polyak well-posedness are considered in the second section.Firstly,we introduce three kinds of pointwise Levitin-Polyak well-posedness of set optimization problems,namely,pointwise LevitinPolyak L-well-posedness,pointwise Levitin-Polyak B-well-posedness and pointwise LevitinPolyak DH-well-posedness.Then the relationship among these three kinds of pointwise Levitin-Polyak well-posedness are given.The relationships between pointwise Levitin-Polyak well-posedness of set optimization problems and Levitin-Polyak well-posedness of a scalar optimization problem are considered in the third section.Firstly,we introduce the notions of Levitin-Polyak well-posedness and Levitin-Polyak well-posedness in the generalized sense of a scalar optimization problem.Then the relationships between pointwise Levitin-Polyak well-posedness of set optimization problems and Levitin-Polyak well-posedness of a scalar optimization problem are obtained.The chapter 4 includes four parts.The first part is the preliminary knowledge required for this chapter.The relationships between generalized l-well-posedness with perturbation and l-well-setness with perturbation of set optimization problems are considered in the second section.Firstly,we introduce the notions of generalized asymptotically minimizing sequence,generalized l-well-posedness with perturbation and l-well-setness with perturbation of set optimization problems.Then we give the relationships between generalized l-well-posedness with perturbation and l-well-setness with perturbation of set optimization problems.The relationships between semicontinuity of l-minimal solution mappings of parametric set optimization problems and the generalized l-well-posedness with perturbation of set optimization problems are considered in the third section.The relationship between generalized l-well-posedness with perturbation of set optimization problems and generalized well-posedness with perturbation of scalar optimization problems is established in the fourth section.In chapter 5,main results of this thesis are briefly summarized.Some problems which are remained and thought over in future are put forward.