Study On Optimality Condition And Duality For Several Classes Of Set-valued Optimization Problems

The optimality conditions for set-valued optimization problems(necessary and sufficient conditions for the existence of efficient solution) and duality theory are the main contents of set-valued optimization theory. Moreover, variational inequality problem plays an important role in the research of existence of solutions for set-valued optimization problem. In this thesis, the existence results of vector set-valued optimization problems and related vector variational-like inequalities are mainly researched, and the dual theory for vector set-valued optimization problems is also studied.1. The development and current research on the topic of the existence of solutions and dual theory for vector optimization problems(specially for set-valued optimization problems)are described. The relationships between vector optimization problems and variational inequality problems are also researched. Some required symbols and definitions are introduced.2. The existences of weak efficient solution for nonconvex and semistrictly quasiconvex set-valued optimization problems are established respectively in Hausdorff topology vector spaces ordered by a convex cone; the properties of weak solutions set are also studied.3. By using the concepts of subdifferential(introduced by Baier and Jahn in 1999) andη-subdifferential of a set-valued mapping, several types of generalized Stampacchia vector variational-like inequality(for short, SVVLI) problems are defined and the solvability of these SVVLI problems are studied. Existence results of weakly efficient solutions for set-valued optimization problems are established by exploiting the existence of solutions for SVVLI problems via a Fan-KKM lemma or a fixed point theorem.4. A general augmented vector-valued function which possesses certain monotonicity is introduced and then used to construct the general conjugate dual problem for a set-valued optimization problem on the base of weak efficiency. A result on strong duality is established between primal problem and dual problem via an abstract subdifferential. By using the augmented vector-valued function which possesses the property of valley-at-0, several types of dual problems for cone-constraint set-valued optimization problems are constructed and the weak and strong dual results are also obtained. Some sufficient conditions for the existence of Lagrangian multiplier of set-valued optimization problem are proved.