(C,?)-Super Subdifferentials Of Set-Valued Maps And Optimality Conditions For Set-Valued Optimization Problems

In optimization theory,the solution of set-valued optimization problem is a very important subject for establishing the optimality condition of set-valued optimization problem.However,some scholars point out that the set of efficient solutions or weak efficient solutions is large and their properties are not good enough.In this paper,we study the(C,?)-super subdifferential of set-valued mappings by using approximate solutions.Firstly,we introduce the concept of(C,?)-super efficient point of a set,and give some properties and equivalent characterizations of(C,?)-super efficient point of a set,and compare it with the efficient points in the existing literature.Furthermore,in the sense of(C,?)-super efficiency,we obtain the scalarization theorem of set-valued optimization problems.Secondly,we define the(C,?)-Super subdifferential of set-valued maps,and study the existence conditions of(C,?)-Super subdifferential of set-valued maps.In addition,we obtain the Moreau-Rockafellar theorem characterized by(C,?)-Super subdifferential of set-valued maps.As an application,we establish optimality conditions for set-valued optimization problems by using(C,?)-super subdifferential of set-valued maps.The main contents of this paper can be summarized as follows:In the first chapter,we first review the background and significance of set-valued optimization problems.Secondly,we review the research status of the solution of optimization problem at home and abroad.Finally,we give the main content of this paper.In the second chapter,we introduce some basic definitions and related theories,including the base of cone,the D-convexity of set-valued mapping,the approximate D-subconvex of set-valued mapping,and the approximate(C,?)-subconvex of set-valued mapping.In the third chapter,we define the(C,?)-efficient point of set M,the(C,?)-weakly efficient point of set M,the(C,?)-Henig proper efficient point of set M,the(C,?)-super efficient point of set M,the(C,?)-strong efficient point of set M and so on,obtain some properties of them,and compare them among several kinds of true(weak)efficient points.In Chapter 4,we give the scalarization of set(C,?)-super efficient points in locally convex spaces.In Chapter 5,firstly,we define(C,?)-Super subgradient of set-valued maps and(C,?)-Super subdifferential of set-valued maps.Secondly,we study the existence conditions of(C,?)-super subdifferential of set-valued mappings.Finally,we obtain the Moreau-Rockafellar theorem characterized by(C,?)-super subdifferential of set-valued mappings.In Chapter 6,we obtain the optimality conditions for set-valued optimization problems by using(C,?)-super subdifferential of set-valued mappings.