The Optimality Condition And Duality Of A Class Of Composite Nonsmooth Multiobjective Programming

Composite optimization problem is a great important class of optimization problem. This is not only because it covers various common multiobjective problems, but it provides a unified framework for studying the convergence behaviour of many algorithms. Moreover, convex composite programming problem is an important and basic class of optimization problem. However, many programs in our real life are nonconvex and multiobjective. So as to meet the demands of solving practical problems, many generalizations to convexity have been made, among which the cone-invexity is an important form. Therefore, it is significant to study the composite nonsmmoth multiobjective programming under such generalized invexity. This paper is concerned with the optimality conditions, (weak) saddle point theorem and duality results for a class of composite nonsmooth multiobjective programs under the cone generalized invexity.The outline of the thesis is as follows.In chapter 1, we introduce the current situation of composite optimization problems and generalized invexity.In chapter 2, we present some preliminaries for the full article.In chapter 3, we focus on the optimality conditions of composite nonsmooth multiobjective programs, where objective function and the constraints are compositions of locally Lipschitz, and Gateaux differentiable and locally Lipschitz functions. First of all, the necessary optimality condition of composite nonsmooth programs is established by generalized alternative theorem and semi-infinite Gordon theorem. Furthermore, a new class of conditions namedη? generalized null space condition is formulated to discuss the sufficient optimality conditions. At last, some examples are given to show the sufficient optimality condition.In chapter 4, the weak saddle point theorem and saddle point theorem are derived for the composite nonsmooth multiobjective programming under the cone generalized invexity.In chapter 5, Mond-Weir,Wolfe and Mixed duality results are established for composite nonsmmoth multiobjective programs by the cone generalized invexity and η?generalized null space condition, which are weak dual theorems,strong dual theorems and restricted converse dual theorems.In chapter 6, we give a summary of this paper and put forward some problems for further study.The innovation of this thesis lies in the 3-th,4-th,5-th chapter.