The Optimality Conditions And Vector-Valued Lagrange Duality Of The Infinite Dimensional Vector Extremum Problems

In this thesis, some topics on the infinite dimensional vector extremum problems are discussed. The concept of generalized subconvexlike map is defined, some important properties of the new concept are discussed in linear space, and the alternative theorem of the map is established. By the alternative theorem, the optimality conditions of vector extremum problems with generalized inequality constraint are established in linear space. In linear topological space, the concept of the gradient of G-differentiable function is introduced. Applying the theorem of alternative of subconvexlike map and other some conclusions, the optimality conditions represented by gradient for the vector extremum problems with generalized inequality constraint are obtained. In ordered linear topological space, the concept of (u,O₂,Y₊)- generalized subconvexlike map is defined, and some important properties of the concept are discussed, and the alternative theorem of the map is established. The optimality conditions of vector extremum problems with generalized inequality constraint are obtained by the theorem in ordered linear topological space. Under the G-differentiable ,using the alternative theorem of generalized subconvexlike map and other some conclusions in locally-convex Hausdorff topological spaces, the optimality conditions of vector extremum problems with set constraint are established. Finally, in linear space, the concepts of vector Fritz-John saddle point and vector Kuhn-Tucker saddle point are defined, the relations among them and weak efficient solution of vector extremum problems with generalized inequality constraint are discussed. At the same time, vector-valued Lagrange duality of vector extremum problems are established, including weak duality, strong duality and converse duality theorem in linear space.