Study On A Class Of Non-smooth Multi-objective Optimization Problems

The multi-objective optimization problem is an important research direction in the field of optimization theory and its application.The research on smooth multi-objective optimiza-tion problems has made great achievements.However,most of the mathematical models in practical problems are nonsmooth.Besides,the generalized convexity of functions plays an important role in the optimization,therefore,the research on nonsmooth multi-objective optimization problems under generalized convexity is a hot topic in recent years.In this the-sis,several kinds of generalized invex functions are introduced,and the nonsmooth multi-objective optimization problems are studied by using the generalized directional derivative,Jacobian and subdifferential tools,the main results may be summarized as follows:In part one,a class of generalized invex functions,called(?,?,?)-invex functions,is firstly introduced,and some specific examples are given to illustrate their existences.Then,we consider the relationships of solutions between two types of vector variational-like in-equalities and multi-objective programming problem under(?,?,?)-invexity.Finally,the existence results for the discussed variational-like inequalities are proposed by using KKM-Fan theorem.In part two,on the basis of invexity and approximate convexity,several extended ap-proximate invex vector-valued functions of higher-order involving generalized Jacobian are firstly introduced,and some specific examples are introduced to illustrate their existences.Then the notions of higher-order(weak)quasi efficiency with respect to a function are pro-posed for a multi-objective programming.Finally,the relationships between the solutions of generalized vector variational-like inequalities in terms of generalized Jacobian,the gen-eralized quasi efficient solutions of nonsmooth multi-objective programming problems and vector critical point are proposed under the introduced generalization of higher-order ap-proximate invexities assumptions.In part three,two kinds of KKT conditions are introduced,and the sufficient optimality conditions,Wolfe type dual and Mond-Weir type dual theory of nonsmooth mul-tiobjective optimization problems are studied under strong invexity of order m,strong pseudo invexity of order m and strong quasi invexity of order m assumptions,respectively.