The Generalized Convexity Of Set-Valued Maps And Set-Valued Optimization

Set-valued optimization theory is one of the main research fields of optimizationtheory and applications. Its theories and methods are widely used in the areasof di?erential inclusion, variational inequality, optimal control, game theory, eco-nomic equilibrium problem, environmental protection, military decision making,etc. The study of this topic involves many disciplines, such as: set-valued analysis,convex analysis, nonlinear functional analysis, nonsmooth analysis, partial order-ing theory, and so on. Thus, the research for this topic has important theoreticalvalue and practical significance.Both optimality condition and the structure theory of solution set of set-valuedoptimization problems are important components in the set-valued optimizationtheory. The optimality condition is an important foundation for developing opti-mization algorithms. And convexity and generalized convexity play a very impor-tant role in the optimization theory. Thus, the research of the generalized convexityof set-valued maps and the optimality conditions of set-valued optimization prob-lems become a hot issue.It is well known that e?cient solution of set-valued optimization problem is so-lution in the sense of non-inferiority with respect to partial order. Therefore, gener-ally speaking, the set of solutions will be large, just as Geo?rion observed, some ofthe solution set have poor properties. Then, there has been e?orts to seek e?cientsolutions which possess nice properties–called proper e?cient solution. The mainproper e?cient solutions are: super e?cient solution, Benson proper e?cient solu-tion and Henig proper e?cient solution. The super e?cient solution possess niceproperties. But, the existence condition of super e?cient solution is very strong.Meanwhile, for discussing the scalarization theorem and the Lagrangian multipliertheorem in the sense of the Benson proper e?ciency, we have to require that theordering cone with a compact or weak-compact base. But, many positive cones ofnormal space have not a compact or weak-compact base. However, Henig propersolution has many desirable properties, its existence condition is weaker than thatof super e?cient solutions and the ordering cone only requires a base. So far, thereare a few papers which deal with Henig proper solution. Then, the thesis mainlystudy the generalized convexity of set-valued mappings and Henig proper e?ciencyfor set-valued optimization.The thesis is divided into seven chapters, the main contents are as follows:In Chapter 1, firstly, we give brief introduction to the development and researchsignificance of set-valued optimization problem. Secondly, the relations amongweak e?cient points, e?cient points and proper e?cient points of a set are given,some concepts and results are presented. Finally, we summarize the developmentof the generalized convexity of set-valued maps and the optimality condition of set-valued optimization problems, the approximate solutions of set-valued optimization problems and the connectedness of the solution set in Section 1.5.We study the nearly cone-subconvexlike set-valued maps in Chapter 2. Firstly,some characterizations of nearly cone-subconvexlike set-valued maps are derived.Secondly, we prove the ic-cone-convexlike set-valued maps is special case of thenearly cone-subconvexlike set-valued maps. Finally, the relations between nearlycone-convexlikeness and ic-cone-convexlikeness are also given.In Chapter 3, we mainly study Henig proper e?ciency for set-valued optimiza-tion problems. Firstly, we prove that two definitions of Henig proper e?cient pointare equivalent, discuss the relation between Henig proper e?cient points and Ben-son proper e?cient points, and present the existence conditions of Henig propere?cient point. Under the assumption of nearly cone-subconvexlikeness, a scalar-ization theorem, a Lagrange multiplier theorem and two saddle theorems for Henigproper e?cient pair in set-valued optimization problem are established in Section3.3, Section 3.4 and Section 3.5, respectively. As an interesting application of theresults in this chapter, we establish a scalarization theorem , a Lagrange multipliertheorem and a saddle theorem for super e?cient pair in set-valued optimization.In Chapter 4, we study Henig proper e?ciency in vector optimization involvingcone-arcwise connected set-valued maps. Firstly, we present some notations andlemmas that are required in the sequel. In Section 4.3, some important propertiesof the cone-arcwise connected set-valued mapping are derived, especially, the globalresult for Henig proper e?cient pair is proven, the unified necessary and su?cientoptimality conditions of Henig proper e?cient pair and strong e?cient pair areobtained by using cone-directed contingent derivative. In Section 4.4, we provethat the set of Henig e?cient solution and the set of super e?cient solution areconnected under objective mappings are cone-arcwise connected set-valued maps.In Chapter 5, firstly, we obtain some properties of the generalized cone-preinvexmapping. Secondly, the necessary and su?cient optimality conditions of Henigproper e?cient pair and super e?cient pair are obtained. Meanwhile, we establishthe closed relationships between proper e?ciency of set-valued optimization andproper e?ciency of vector variational-like inequalities.In Chapter 6, we study the approximate solutions for vector optimization prob-lem with set-valued maps. Firstly, we present a useful form of Tammer functionand an important characterization of quasiconvex set-valued map in Section 6.2.Secondly, the scalar characterization of approximate solutions is derive withoutimposing any convexity assumption on objective functions in Section 6.3. The re-lations between approximate solutions and weak e?cient solutions are discussed inSection 6.4. Finally, we prove that the set of approximate solutions is connectedunder objective functions are quasiconvex set-valued maps.In Chapter 7, we summarize the main results of the thesis, and put forwardsome issues to be studied.