The Research On Efficiency Of Set-Valued Optimization

The efficiency of the vector optimization problem with set-valued maps in the norm linear space was studied in this paper.The definition and the related properties of sequences of several types about sets were discussed.The related properties of extensions of Gerstewitz's functions were studied by introducing two types of extended real-valued functions defined on ((3)× ((3).The conditions of the convergence and stability of the sequence of the efficient set,weakly efficient set and Henig efficient set were given in the vector optimization problem by using Kuratowski-Painlevéconvergence of the sequence of sets.The main contents were as follows:The chapter one,the vector optimization and its application and the development background and the research status of the vector optimization theory with set-valued maps were introduced.The chapter two,Preliminaries of this article and definitions and the related properties of the sequence of several types were given.The chapter three,Gerstewitz's functions and its related properties of the sequence of several types categories were studied and extended to the more general situation.Based on giving the concept and the related properties of Gerstewitz's functions,the related properties of extensions of Gerstewitz's functions were studied by introducing two types of extended real-valued functions defined on ((3)× ((3)and using the sequence of several types discussed in chapter two.The chapter four,the efficiency of the vector optimization problem with set-valued maps was discussed by using characteristics and properties of different sequences and combining with contents of chapter 1-2.The optimality,stability,existence and other related content of the the efficiency of the vector optimization problem with set-valued maps were discussed.Finally,the stability of the sequence of efficient sets for a kind of proper quasi-convex cone mapping vector optimization problem was studied.