Study On Optimality For Multiobective Programming Problem With Generalized Subdifferential

The study on the optimality of multiobjective programming problems has great the-oretical significance. So far, there has had amounts of achievement in the study of the differentiable multiobjective programming problem’s optimality. However, the objective and the constraint functions might be nondifferentiable sometimes. Then it is meaningful to study nonsmooth optimization problems through some generalized differential tools. The classical generalized differential tools include the Clarke subdifferential, the Mor-dukhovich’s subdifferential and so on. In this paper, we mainly use the Mordukhovich’s subdifferential to investigate the optimality of the nonsmooth multiobjective programming problem with inequlity constraints. Some regularity conditions of the multiobjective pro-gramming problem and their relations are discussed. What’s more, we established a weak Kuhn-Tucker necessary optimality condition for the Geoffrion proper efficient solution of the multiobjective programming problem. At the same time, the necessary and sufficient optimality condition of the weak efficiency solutions and the sufficient optimality condi-tion of the efficiency solutions for the multiobjective programming problems are given by the tools such as the pseudoconvex in the term of the Mordukhovich’s subdifferential.In Chapter 1, we introduce the research background, research meaning and the re-search status. Especially, we review the study of functions’ generalized convexity, the generalized differentials and the regularity conditions for the multiobjective programming problems.In Chapter 2, we mainly introduce a kind of the linear cones in the term of the Mordukhovich’s subdifferential. Then we discuss the relationships between them and the linear cones which are generated by the Clarke derivative. Through these linear cones we establish some regularity conditions in the sense of the Mordukhovich’s subdifferential including (EGARCM), (GARCM), (GCRCM), and (GGRCM). At last, the rela-tions among them are obtained, so are the relations between them and the corresponding regularity conditions in the sense of the Clarke derivative.In Chapter 3, we first obtain a weak Kuhn-Tucker necessary optimality condition for the Geoffrion proper efficient solution with the regularity conditions which are given in Chapter 2. Secondly, the necessary and sufficient optimality condition of the weak efficiency solutions and the sufficient optimality condition of the efficiency solutions are established with the pseudoconvex in the term of the Mordukhovich’s subdifferential. Thirdly, an equivalent description of the efficient solutions’ sufficient optimality condition are proposed with the linear cones in the sense of the Mordukhovich’s subdifferential.