The Solutions And Duality Of Vector Optimization Problems

Vector optimization is one of the main research fields of optimization theory and pplications. The study of this topic involves many disciplines, such as:convex analysis, ariational analysis, nonlinear analysis, and so on. The theory and methods of vector op-imization are widely used in many areas of engineering design, economic management, nvironmental protection, health and medical community, transportation, etc. Therefore, he research on this topic has both theoretical value and practical significance. In this hesis, we mainly concentrate on the solutions and duality of vector optimization in three ispects:characterizations of approximate solutions and scalarization in vector optimiza-ion problems; second order duality of vector optimization problems; some properties generalized convexity for vector-valued mappings and their applications in vector >ptimization problems. The whole thesis is divided into seven chapters, and the main contents are as follows:In Chapter 1, we give a brief introduction about the background and significance of vector optimization theory and applications, and also summarize the historical develop-ment of vector optimization in some aspects associated with this thesis. Then we recall some basic concepts and theory which will be used in the thesis. Finally, we introduce the motivation and the main contents of this thesis.Chapter 2 devotes to characterizations of approximate solutions in vector optimiza-tion problem via linear scalarization in a separated locally convex space. Firstly, we consider some dual properties of free disposal sets, and establish the dual relationship between the intersection of free disposal sets and the addition of dual sets and the dual relation between the addition of free disposal sets and the intersection of dual sets, re-spectively. Then, in terms of linear scalarization scheme, we characterize approximate solutions of vector optimization problem, which are defined by free disposal sets and co-radiant sets, respectively. We mainly present necessary and sufficient conditions such that approximate weakly efficient solution set and approximate properly efficient solu-tion set of vector optimization problem can be expressed as the union of the approximate solution sets of the corresponding linear scalarization problems, and also show that many previously corresponding results appear as special cases. Meanwhile, we give some ex-amples to illustrate that such characterization does not hold for approximate efficient solution set of vector optimization problem.In Chapter 3, we focus on nonlinear scalarization for characterizations of approxi-mate solutions of vector optimization problem in real topological vector space. Firstly, we consider the applications of the nonlinear scalarization function of Gerstewitz func-tional in vector optimization. (C,ε)-weakly efficient solutions and (C,ε)-efficient solu-tions of vector optimization problem, which are defined by co-radiant sets, are character-ized via Gerstewitz functional, and the range of approximation of scalarization problem is estimated. Then, we give some shows on recent results about the characterization of (C,ε)-properly efficient solutions in vector optimization problem, which are obtained in terms of nonlinear scalarization function of A function, and also give some examples to illustrate the main results.In Chapter 4, we consider the duality for a special class of vector optimization problem, namely, second order duality for a class of multiobjective optimization prob-lem with cone constraints. Firstly, four types of second order dual models are formulated on the base of second-order approximation. Then, four weak duality theorems are de-rived by means of second-order generalized convexity, and strong duality theorems are established in terms of the generalized Fritz John type necessary optimality condition-s and second-order generalized convexity. Furthermore, based on weak duality results, converse duality theorems are presented under some mild assumptions.In Chapter 5, we investigate some new properties of semi-preinvexity for vector-valued mappings in the sense of cone. We firstly obtain an important property of the vector-valued function η with respect to the second variable. Then, combined with the result of density, we characterize D-semi-preinvexity via D-semi-strictly semi-prequasiinvexity and D-strictly semi-prequasiinvexity, recpectively. Also, a characteri-zation of D-semi-preinvexity is derived via D-semi-prequasiinvexity.In Chapter 6, a class of nonsmooth vector optimization problem with constraints is considered. In terms of Clarke’s subdifferential, two concepts of generalized convexity for vector-valued mapping, named as FJ-pseudoinvexity-Ⅰ(Ⅱ), are introduced in the sense of cone. By establishing Gordan’s theorem over general cone domains, we characterize FJ-pseudoinvexity-Ⅰ(Ⅱ) and obtain equivalent relationships between FJ vector critical points and (weak) efficient solutions of vector optimization problem.In Chapter 7, we summarize the main results of this thesis, and put forward some questions for further study.