Study On The Properties Of E-convex Functions And ?-convex Functions

Convexity plays an irreplaceable role both in pure mathematics such as optimization theory and equilibrium problem,and in engineering,economy,management,and even the national defense.The status of convexity is self-evident.There are plenty of non-convex situations in both real life and corporate production,so the exploration of generalized convexity becomes significant.This paper focuses on the properties of the generalized convexities which are E-convex functions and ?-convex functions.There are two main parts for the discussion of E-convex functions:1,the study of E-convex single objective programming,the properties of the optimal solution set and its equivalent characterization are obtained by the Gateaux differentiable and E-subdifferential of E-convex functions and E-feasible direction of E-convex set.2,the study of E-pseudo-invex multi-objective programming,by making use of hypothesis A' and hypothesis C',is proposed for the equivalent characterization of its local efficient solution and local weakly efficient solution.In addition,the definition of ?-subdifferential for ?-convex functions is given,and some basic properties are listed here.At the end of this paper,a brief equivalent characterization is given for the ?-convex functions.