Study On Optimality Conditions Of Generalized Strong Convex Multi-objective Optimization

As we all know,strong convexity plays an important role in the study of optimization theory and application.The following three parts will be the main content of our study.The first part,relationships between vector variational inequality and non-smooth multiobjective optimization.We first introduce an extension of higher-order strong pseudoconvexity for Lipschitz functions,termed higher-order strongly pseudo-convex functions of type-I,and some examples are presented in the support of this generalization.Then,we identify the strict minimizers of higher order,the vector critical points and the solutions of the weak vector variation inequality problem under the higher-order strong pseudo-convexity of type I hypothesis.The second part,optimality conditions for the constrained non-smooth multi-objective optimization problem.We first introduce the notion of higher order strong pseudo-convexity of type-I for Lipschitz functions,and some examples are presented to illustrate their existences.Secondly,it presents the local-global property and an optimality sufficient condition for the strict minimizer of higher order to a non-smooth multi-objective optimization problem with inequalities constraints,we propose a generalized Guignard constraint qualification and a generalized Abadie constraint qualification for this problem under which necessary optimality conditions are proved.Finally,we define mixed saddle point of order mfor a partial vector-valued Lagrangian of a multi-objective optimization problem.The equivalence of these saddle points and the higher-order strict minimizers for(CNMOP)is established under higher order strong pseudo-convexity of type-I conditions involved.The last part,h-strongly convex set-valued mappings in inner product spaces.A kind of generalized strongly convex set-valued mappings,termed h-strongly convex set-valued mappings,is introduced in real normed spaces.Then,by employing R?adstr ¨om cancellation law,some basic properties of h-strongly convex set-valued mappings are proposed.Finally,a characterization of inner product spaces involving the h-strongly convex set-valued mapping is presented.