Benson, Efficient Solution Of Significance Set-valued Mapping Generalized Gradient And Set-valued Optimization

This paper defines the generalized gradient of set-valued mappings with Benson proper efficiency significance, and to study some applications of generalized gradient in set-valued optimization.This paper consists of four chapters.The first chapter is an introduction, which is a brief introduction of the article's background.In the second chapter, in real normed linear spaces, by use of the generalized contingent derivatives of set-valued mappings with Benson proper efficiency significance, a kind of generalized gradient of set-valued mappings is introduced, its existence is shown by the separation theorem of convex sets .In chapter 3, on the basis of the second chapter, some basic properties of the generalized gradient of set-valued mappings under Benson proper efficiency significance are discussed. We study the convexity of the generalized contingent derivatives, show an equivalent theorem of the generalized gradient, and prove the Moreau-Rockafellar theorem of the generalized gradient of set-valued mappings under Benson proper efficiency significance.In chapter 4, we introduce some applications of the generalized gradient of set-valued mappings under Benson proper efficiency significance in set-valued optimization.