Some Applications Of Higher-Order Derivatives For Set-Valued Mappings In Vector Optimization

In this thesis, we study higher-order optimality conditions for set-valued optimization problems, higher-order optimality conditions for nonconvex set-valued optimization problems, and higher-order Mond-Weir and Wolfe type duality problems for constrained set-valued optimization problems. We also study higher-order sensitivity and second-order stability for vector optimization problems. This thesis is divided into eight chapters. It is organized as follows:In Chapter 1, we describe the development and current researches on the topic of optimality conditions and duality for set-valued vector optimization problems, and stability and sensitivity for vector optimization problems. We also give the motivation and the main research works.In Chapter 2,we introduce some basic assumptions and concepts in this thesis. We introduce generalized higher-order contingent sets and generalized higher-order adjacent sets for sets, and also discuss some properties of them.In Chapter 3, we introduce generalized higher-order contingent epiderivatives and generalized higher-order adjacent epiderivatives of set-valued maps, and discuss some important properties of the two kinds of higher-order derivatives. By virtue of generalized higher-order epiderivatives and Henig efficiency, we obtain higher-order necessary and sufficient optimality conditions for Henig efficient solutions to a set-valued optimization problem whose constraint set is determined by a fixed set. Simultaneously, we also obtain higher-order Kuhn-Tucker type necessary and sufficient optimality conditions for Henig efficient solutions to a set-valued optimization problem whose constraint set is determined by a set-valued map.In Chapter 4, we introduce higher-order generalized contingent derivatives and higher-order generalized adjacent derivatives for set-valued maps, and discuss some of their properties. By virtue of these properties and some properties of generalized higher-order epiderivatives in Chapter 3, without any convexity assumptions, we obtain necessary and sufficient optimality conditions of weak efficient solutions for unconstrained set-valued optimization problems and constrained set-valued optimization problems, respectively.In Chapter 5, we introduce higher-order weakly generalized contingent epiderivatives and higher-order weakly generalized adjacent epiderivatives of set-valued maps,and discuss some of their properties. By virtue of higher-order weakly generalized adjacent epiderivatives,we introduce a higher-order Mond-Weir type dual problem and a higher-order Wolfe type dual problem for a constrained set-valued optimization problem, and discuss the corresponding weak duality, strong duality and converse duality properties, respectively.In Chapter 6, we study higher-order sensitivity in vector optimization problem. Firstly, we discuss relationships between higher-order contingent derivatives (higher-order adjacent derivatives) of set-valued maps and their profile maps. Secondly, given a family of parametrized vector optimization problems, we define a perturbation map and a weak perturbation map for the problems, respectively. Finally, we investigate the relationship between higher-order adjacent derivatives for the weak perturbation maps and weakly minimal point sets for higher-order adjacent derivatives of feasible set maps in the objective space. We also investigate the relationships between higher-order adjacent derivatives for the perturbation maps and two types of minimal point sets (i.e. minimal point sets and weakly minimal point sets) for higher-order adjacent derivatives of feasible set maps in the objective space.In Chapter 7, we study stability of second-order derivatives in vector optimization problems. Firstly, we establish continuity and closedness of second-order contingent derivatives and second-order adjacent derivatives for set-valued maps. Secondly, given a family of parametrized vector optimization problems, we define the weak perturbation maps for the problems. Finally, under suitable assumptions, we obtain the upper semicontinuity and the lower semicontinuity of second-order adjacent derivatives for the weak perturbation maps.In Chapter 8,we summarize the results of this thesis and make some discussions.