Characterization Of Solution Sets Of E-convex Programming Problems

The theory of convexity play a central role in mathematical economics,management science, engineering, and optimization theory. In this paper we mainlystudy a class of important generalized convexity, E-Convexity. According to thedefinition of E-subdifferential and E-direction derivative, first we propose the notionof E-Gateaux differential of E-convex function, get some characteristic theorems ofthe E-Gateaux differential and the E-subdifferential of E-convex function, then wepropose the equivalent characterizations of the solution sets of E-convexprogramming problems by using the characteristic theorems, For an E-convexprogram in a normed vector space with the objective function admitting theE-Gateaux differential at an optimal solution, we show that the solution set consistsof the feasible points lying in the hyperplane whose normal vector equals theE-Gateaux differential.This paper is divided into six chapters,the outline is as follows:In chapter1, we introduce the significance and the current situation ofE-Convexity and the characterization of solution sets.In chapter2, we present some preliminaries for the full article.In chapter3, we establish the characteristic theorem of the E-Gateauxdifferential and the E-subdifferential, Prove the equivalence of two subdifferentialdefined in this article, get the monotonicity of E-subdifferential, study thecharacteristic relationship between the E-Gateaux differential and E-subdifferential,and the characteristic relationship between the E-Gateaux differential and thesolution of of E-convex programming problems. In chapter4, In the conditions of the E-Gateaux differentiable, the equivalentcharacterizations of the solution sets of E-convex programming problems are derivedby using the characteristic theorem of E-Gateaux differential.In chapter5, we promote the conclusions of Chapter4in the conditions of theE-subdifferential is nonempty, get the equivalent characterizations of the solutionsets of E-convex programming problems.In chapter6, we give a summary of this paper and put forward some problemsfor further study.The innovation of this thesis lies in the3-th,4-th,5-th chapter.