In this paper, with the aid of idea of defining those sets and functions, the definitions of E -convex cone, pseudo-quasi- E -convex functions, E -direction derivative are discussed.

This paper is intended to discuss the properties of-convex sets by introducing definition of convex sets,definite-convex combination,-convex inclusion and-convex cone and applying definition of-convex and its properties and obtaining some useful results.

General convex sets and convex functions: E-convex sets and E-convex functions are investigated, some properties of which are achieved, the ctiterions of E-convex functions and quasi-semi-E-convex functions are given, minimizing sequences and critical sequences in optimization problems are studied.

Strong E-convex set,strong E-convex function and strong E-convex programming;

On level sets of E-convex function and E-quasiconvex function

Some new properties of their E-convex functions are discussed on the basis of the former researches,and their applications in optimization are researched,some of established Conclusions expanded,these functions perfected.

Recently a new criterion of quasi-semi-E-convex functions was introduced by Peng in 2006 for a new criterion of quisi-semi-E-convex functions.

And a counter example about a result of E-convex programming is given.