Abstract: In this paper, an important class of generalized convex programming problems, E-convex program, was considered. We defined the E-Gateaux differential of E-convex function on the E-convex set,  and got some characteristic theorems  of the E-Gateaux differential of E-convex function, proposed the equivalent characterizations of the solution sets of E-convex programming problems by using the characteristic theorems. For an E-convex program in a normed vector space with the objective function admitting the E-Gateaux differential at an optimal solution, we showed that the solution set consists of the feasible points lying in the hyperplane whose normal vector equals the E-Gateaux differential.

Key words: E-gateaux differential, characterization of solution sets, E-convex function, E-convex program, sub-differential