Higher-order Optimality Conditions And Dual Theory For Set-valued Optimization Problems

The optimality condition and dual theory of set-valued optimization problems are important parts of set-valued optimization theory.In this paper,the higherorder optimality conditions of a unconstrained composite set-valued optimization problem and the higher-order dual theory of a constrained set-valued optimization problem are studied.1.The higher-order generalized weak Studniarski epiderivative and the higherorder weakly adjacent epiderivative of set-valued maps without lower-order approximation directions are introduced,and some of their properties are discussed.Firstly,the higher-order adjacent set without low-order approximation directions are defined,and some properties of the higher-order adjacent set and the higher-order Studniarsk set are established.Then,two new types of generalized cone-convex setvalued maps are proposed,which generalize cone-convex set-valued maps.Finally,with the help of the higher-order Studniarsk set and weak efficiency,we introduce a higher-order derivative which is better than the higher-order generalized Studniarski epiderivative of set-valued maps in reference [21],that is,the higher-order generalized weak Studniarski epiderivative of set-valued maps,and obtain some properties of the higher-order generalized weak Studniarski epiderivative,chain operation rules and sum operation rules.At the same time,by using weak efficiency and the higherorder adjacent set,a higher-order derivative which is better than the higher-order derivative of set-valued maps in reference [52] is introduced,that is,the higherorder weakly adjacent epiderivative of set-valued maps,and some properties of the higher-order weakly adjacent epiderivative are established.2.The model of unconstrained composite set-valued optimization problem is established.Based on the higher-order generalized weak Studniarski epiderivative and its properties,the higher-order optimality sufficient and necessary conditions for the weakly efficient solution of the problem are established under the generalized cone-convex hypothesies,and some examples are given to show the results obtained.3.By using the higher-order weakly adjacent epiderivative,the higher-order Mond-Weir type dual model and Wolfe type dual model for a constrained set-valued optimization problem are constructed,and the corresponding weak duality,strong duality and inverse duality theorems are established respectively under the generalized cone-convex hypothesis,which improve the corresponding results in reference[52].