Study On Scalarization Of Approximate Solutions Of Vector Optimization Problems

Study on characterizations of approximate solutions of vector optimization problems is an important research aspect in the field of the theory and methods of vector optimization problems. Scalarization methods are important methods to study the characterizations of solutions for vector optimization problems, mainly including as linear scalarization method based on generalized convexity and the corresponding alternative theorem, and nonlinear method via nonlinear scalarization functions and the corresponding nonconvex separation theorems. This paper mainly focuses on some characterizations of approximate solutions of vector optimization problems by means of improvement sets, generalized interior, gener-alized convexity and two kinds of the classical nonlinear scalarization functions, including as some topological properties and dual characterizations of improvement sets, some gener-alized interior characterizations via quasi interior for improvement sets, linear scalarization characterizations of weakly E-efficient solutions of vector optimization problems, and some nonlinear scalarization characterizations of e-properly efficient solutions and E-Benson proper efficient solutions of vector optimization problems under the condition of emptiness of the topological interior of the ordering cone.Chapter 1 mainly gives some research advancements about characterizations of solutions for vector optimization problems and some basic concepts.Chapter 2 mainly focuses on topological properties and dual properties of improvement sets. Firstly, some operation properties of improvement sets are given with nonempty topo-logical interior. Furthermore, by using improvement sets, a strong version of Assumption B proposed by Flores-Bazan and Hernandez is proposed. Based on some classical dual characterizations of convex cone and by means of some tools such as separation theorem for convex sets and recession cone, some dual characterizations of improvement sets are given.Chapter 3 mainly focuses on some properties of improvement sets and linear scalar-ization characterizations of weakly E-efficient solutions in the sense of quasi interior for set-valued vector optimization problems. Firstly, some properties of improvement sets are given via quasi interior. As applications, the corresponding alternative theorem is established by means of improvement sets and quasi interior, and linear scalarization results of weakly E-efficient solutions are obtained for set-valued vector optimization problems.Chapter 4 mainly focuses on some nonlinear scalarization characterizations of ε-properly efficient solutions of vector optimization problems. By means of Gerstewitz nonlinear scalarization functions and the corresponding nonconvex separation theorem, some nonlinear scalarization characterizations of ε-properly efficient solutions are estab-lished for vector optimization problems.Chapter 5 mainly focuses on some nonlinear scalarization characterizations of E-Benson proper efficient solutions. By means of a kind of nonlinear scalarization function named as △ function defined in normed linear space and the corresponding nonconvex separation theorem, some nonlinear scalarization results of E-Benson proper efficient solutions are established for vector optimization problems.