The Optimality Conditions With Set-valued Optimization

The theory of vector optimization with set-valued maps finds wide applications in differential inclusions, approximation theory, variations, optimization control, and so on, the optimality conditions for set-valued optimization problems are its important components and are the important base of developing modem algorithms. On the other hand, the concept of convexity plays important roles in the Optimization theory,hence each of generalizations of convexity receiveser searcher's attentions. The thesis is to gain the optimality conditions are obtained for set-valued vector optimization problems in a sense of weak efficient solutions by applying the alternative theorem, to obtain the optimality conditions with without derivatives for set-valued optimization problem in the sense of superly eficient element of normed linear space,and to obtain the optimality conditions under the assumption of nearly cone-subconvexlike set-valued functions. For details,these results are given in the following.1. the optimality conditions of a class of set-valued optimization problem and weak efficient solutions and vector saddle points of set-valued optimization problems are investigated in linear spaces. under the assumption of generalized -subconvexlike, Kuhn-Tucker necessary condition and sufficient condition of a class of set-valued vector optimization probles are established by applying the alternative theorem;in ordered linear spaces,generalized vector fritz-john saddle point and generalized vetor kuhn-tucker saddle point of set-valued optimization problems with generalized inequality constraints are defined ,and the relations between them are established. and then, applying the alternative theorems under the generalized cone-subconvexlikeness,the relations among the weak efficient solutions of set-valued optimization problems and them are dicussed.2. under the relative interior, the optimality conditions of weak efficient solutions for the optimization problem of a class of generalized convex set-valued mappiong and to obtain the optimality conditions for the optimization problem with generalized equality and inequality constraints under the assumption of nearly cone-subconvexlikeset-valued functions in linear topological space. under the relative interior, Kuhn-Tucker necessary condition and sufficient condition for the optimization problem of a class of generalized convex set-valued mappiong by applying the alternative theorem in a sense of weak efficient solutions; under the assumption of nearly cone-subconvexlikeness, to obtain Kuhn-Tucker necessary condition and sufficient condition for the optimization problem with generalized equality and inequality constraints by applying the alternative theorem in a sense of weak efficient solutions .3. The super efficiency of normed linear space for set-valued optimization problem is investigate.Under the assumption of Generalized subconvexlikeness, by the alternative theorem,scalarization theorem of set-valued vector optimization problems with equality and inequality constraint in a sense of super efficient solutions,Finally, Kuhn-Tucker necessary condition and sufficient condition of set-valued vector optimization proble are established.