Optimality Conditions Of Set-valued Optimization

Vector optimization is an important branch of optimization theory, and set-valued optimization, which is widely applied in many areas such as mathemati-cal programming,nonsmooth analysis,mathematical economics,engineering and management, is an important part of vector optimization. Recently, many scholars have been interested in it. We note that the topological interior of the ordered cone is an important notation when we study optimization problems, but, how do we establish optimality conditions when the topological interior of the ordered cone is empty? We also note that convexity plays a very important role in the optimality conditions for optimality problems, however, we find that the conditions with con-vexity are not satisfied in some optimization problems. Hence, it is very necessary to generalize the usual topological interior of the ordered cone and weaken the con-vexity of the function. It is well-known that it is very difficult to find "solution" of vector optimization problem. Therefore, it is very meaningful to establish the optimality conditions of the vector optimization problems in the sense of different solutions. In this thesis, making use of different kinds of weakened interiors of the ordered cone, a series of optimality conditions for set-valued optimality problems are established in the sense of different generalized convexity and efficiency. This thesis is dived into five chapters, and the main contents are as follows.In chapter one, firstly, wo recall the concepts of generalized convex set-valued maps. Secondly, we recall the progress of the theorems of the alternative and optimality conditions. Thirdly, we recall efficiency and duality theory of vector set-valued optimization problems. Finally, we give the motivation of this thesis and the main contents studied.In chapter two. firstly, making use of vector closure, we define nearly cone-subconvexlike set-valued map in ordered linear space, and five equivalent propo-sitions about nearly cone-subconvexlike set-valued map and a property that the partial scalarization of nearly cone-subconvexlike set-valued map is also nearly cone-subconvexlike set-valued map in product space are obtained. Secondly, in the sense of Benson proper efficiency, we give the relation between the Benson properly efficient solution of nearly cone-suboonvexlike set-valued optimization problem and the optimality.solution of scalarized problem, and we obtain Lagrange multipliers rule and saddle point theorem of set-valued optimization problems. Thirdly, in or-dered linear space, we obtain some properties of relative algebraic interior, and the properties of generalized cone-subconvexlike set-valued maps defined by the relative algebraic interior are studied. Finally, we obtain a separation property of gener-alized cone-subconvexlike set-valued maps based on the relative algebraic interior, and Knhn-Tncker necessary condition is established by the separation property.In chapter three, we study set-valued optimization problems in separated locally convex spaces. Firstly, we obtain some properties of generalized cone-subconvexlike set-valued map characterized by the relative topological interior. Secondly, by sepa-ration property of generalized cone-subconvexlike set-valued map based on the rela-tive topological interior, we obtain Kuhn-Tucker sufficient and necessary conditions and a scalarization theorem. Thirdly, we use the quasi-relative interior to define generalized cone-subconvexlike set-valued map, and the conditions of separation theorem involving the quasi-relative interior are analyzed, and a separation prop-erty with generalized cone-subconvexlike set-valued map based on the quasi-relative interior is obtained. Finally, using the separation property obtained, we established a series of optimality conditions, including Kuhn-Tuckcr condition scalarization theorem,saddle theorem and duality theorem.In chapter four, we study approximatly strict subdifferential of set-valued map and optimality conditions of set-valued optimization problems. Firstly, we obtain the existing condition and property of approximatly strict subdifferential with set-valued map. Secondly, we obtain a generalized Moreau-Rockafellar theorem char-acterized by approximatly strict subdifferential. Finally, under the assumption of nearly cone-subconvexlike set-valued map, we obtained optimality conditions characterized by approximatly strict subdifferential.In chapter five, under the assumption of strongly G-preinvex function, we study optimality condition of vector-valued optimization problem. Firstly, we give the relations between strongly G-preinvex function and G-preinvex function, strictly G-preinvex function and semistrictly G-preinvex function. Finally, under the as-sumption of strongly G-preinvex function, we obtain that m order local minimizer is m order global minimizer for a vector optimization.