Some Research On Optimality And Duality With Image Space Analysis

In this thesis,some optimality conditions and duality theory of constrained optimization problems from the perspective of Image Space Analysis are investigated.Lagrangian type sufficient optimality conditions and Karush-Kuhn-Tucker necessary conditions on(weak)efficient solutions as well as saddle point type sufficient optimality conditions and several necessary optimality conditions on E-optimal solutions are proved for vector optimization problems.The duality theory of a generalized quasi-equilibrium problem follows the idea of Lagrange duality and the conjugate duality theory of vector optimization problems are established.The thesis is divided into seven chapters and organized as follows:In Chapter 1,the research background and recent developments of optimization problems are firstly recalled.Then basic features of Image Space Analysis,various kinds of solutions for vector optimization problems,related researches on sufficient and necessary optimality conditions as well as duality theory in scalar and vector optimazation problems are reviewed.At last,the motivations and the main research work of this thesis are introduced.In Chapter 2,some concepts,basic notations,several important functions including indicator function,the distance function,Gerstwitz function and the oriented distance function,and certain definitions on tangent cones,normal cone,directional derivatives,subdifferentials and Lipschitz continuty are stated,which will be used in the following context.In Chapter 3,weak separation functions in the Image Space for general constrained vector optimization problems on efficient solutions and weak efficient solutions are proposed.Gerstewitz function is applied to construct a special class of nonlinear separation functions as well as the corresponding generalized Lagrangian functions.By virtue of such nonlinear separation functions,we derive Lagrangian type sufficient optimality conditions in a general context.Especially for nonconvex problems,we establish Lagrangian-type necessary optimality conditions under suitable restriction conditions,and we further deduce Karush-Kuhn-Tucker necessary conditions in terms of Clarke subdifferentials.Finally,we consider linear multi-objective optimization problems as an application to give an equivalent characterization for the set of efficient solutions.In Chapter 4,some optimality conditions in terms of E-optimal solution for constrained multi-objective optimization problems in a general scheme are considered,where E is an improvement set with respect to a nontrivial closed convex point cone with apex at the origin.In the case where E is not convex,nonlinear vector regular weak separation functions and scalar weak separation functions are introduced respectively to realize the separation between the two suitable sets in the Image Space,and saddle point type sufficient optimality conditions are established.In addition,we construct the strong separation functions for E-optimal solution of such a problem and derive the corresponding necessary optimality conditions via the theorem of strong alternative.In Chapter 5,the duality theory of a generalized quasi-equilibrium problem is investigated by using Image Space Analysis approach.Generalized quasi-equilibrium problem is first transformed into a minimization problem.The minimization problem is further reformulated as an image problem by virtue of linear/nonlinear separation function.The dual problem of the image problem is constructed in the image space,then zero duality gap between the image problem and its dual problem is proved to hold under saddle point condition as well as the equivalent linear/nonlinear regular separation condition.Finally,more sufficient conditions guaranteeing zero duality gap are also proposed.In Chapter 6,conjugate duality theory for general constrained vector optimization problems are studied.We introduce the concepts of conjugate map and subdifferential by using two types of maximums.We also construct the conjugate duality problems via perturbation method.Moreover,the separation condition is proposed by means of vector weak separation functions.Then,it is proved to be a new sufficient condition,which ensures the strong duality theorem.In Chapter 7,a brief summary of the main results of this thesis is stated.Some problems which deserve further investigation and exploration are put forward.