Study On Scalarization Via Generalized Interior In Vector Optimization Problems

In recent years,the research of vector optimization theory and method has drawn extensive attention from researchers at home and abroad.The study of the properties of solution to vector optimization problems,especially the study of approximate solutions,has become a very important research direction in the field of vector optimization theory and methods.Scalarization methods are important methods to study the characterizations of solutions for vector optimization problems,mainly including linear Scalarization method based on generalized convexity hypothesis and the corresponding alternative theorem,and non-linear Scalarization method in view of nonlinear scalarization functions and the corresponding nonconvex separation theorems.This paper mainly focuses on some characterizations of approximate solutions of vector optimization problems via improvement sets,generalized interior,and the classical nonlinear scalarization functions,including as some quasi internal properties of the Gerstewitz nonlinear scalarization function,Some generalized interior properties of the free disposal sets within the relative algebraic interior and relative topological interior of the ordered cone and the linear scalarization properties of(weak)efficient solutions for set-valued vector optimization problems.Chapter 1 mainly gives some research advancements for vector optimization problems,mainly including the properties of a variety of exact solutions and various approximate solutions of vector optimization problems,some basic concepts of the generalized interior and some main progress in vector optimizationChapter 2 mainly focuses on quasi interior properties of Gerstewitz nonlinear scalar-ization function.Firstly,under the condition of non-emptiness of quasi interior of the ordering,some new properties are given for the Gerstewitz nonlinear scalarization function and the fact that some results via non-emptiness of topological interior of the ordering cone can not be generalized to the quasi interior case is pointed for the Gerstewitz nonlinear scalarization function.As applications,nonlinear scalarization result of efficient point is established for vector optimization problems.Chapter 3 mainly focuses on some generalized interior properties of the free disposal sets are studied via the relative algebraic interior and relative topological interior of the ordered cone.Furthermore,some linear scalarization results of the corresponding weakly efficient solutions are obtained for vector optimization problems under some suitable conditions.In particular,the main results improve the recent result established by Adan and Novo.Moreover,some concrete examples are also given to illustrate the main results.