Optimality Conditions And Duality For The Multiobjective Programming Involving Generalized Symmetric Invex Function

Multi-objective optimization is a major branch of the optimization category.At the same time,convex function and generalized convex function are a theoretical basis of mathematical programming.Almost all results of multi-objective optimization problems depend on some convexity of objective function and constraint function.Therefore,the convexity and generalized convexity of the function have always been an important topic that people pay close attention to and study in depth.The optimality condition for the existence of weak efficient solutions is an important basis for establishing optimization algorithms.Both it and the dual theory problem of multi-objective programming are hot topics in the field of multi-objective optimization.This paper aims to study the optimality conditions and duality of multi-objective programming problems based on two new generalized convex functions.The main contents are as follows:Firstly,the new generalized convex function is defined by the concept of Minch symmetric gradient,and the convexity is proved by an example.Secondly,under the condition of convexity of this kind,the multi-objective programming problem with support function is studied.Several optimality sufficient conditions are given for the objective function and the constraint are new generalized convex functions.Finally,the original model of Wolfe type dual model and Mond-Weir type dual model are established,and several weak dual theorems,strong dual theorem and strict inverse duality theorem are obtained.On the basis of(V,?)-I symmetric invariant convex functions,the concept of generalized uniform(V,?)-I symmetric invariant convex function is proposed.Which is generalized uniform(V,?)-I type.Symmetric invariant convex function,generalized uniform(V,?)-I-symmetric strictly quasi-invariant convex function,generalized uniform(V,?)-I-symmetric strictly pseudo-invariant convex function and generalized uniformity(V,?)-Type I symmetric strictly pseudo-invariant convex function.Under the new generalized convexity hypothesis,several optimality sufficient conditions for multi-objective programming are proved.At the same time,some dual conclusions of the duality of Wolfe type and Mond-Weir type are studied.In this paper,several new generalized convex functions are proposed,and the optimality conditions and duality of multi-objective programming are studied under the new generalized convexity.The obtained results theoretically extend the existing convexity and enrich the generalized convexity and related theories of multi-objective programming.