E凸规划问题解集的刻画

Characterization of solution sets of E-convex programming problems

考虑一类重要的广义凸规划问题E凸规划,在E凸集中定义了关于E凸函数的E-Gateaux微分概念,证明了E凸函数的E-Gateaux微分的几个特征性质,并利用这些特征性质,提出了E凸规划问题解集的等价刻画,在赋范向量空间中,对于一个目标函数在最优解处E-Gateaux可微的E凸规划问题而言,它的解集是由位于超平面内的可行解组成的,这些可行解的法向量就是目标函数在给定最优解处的E-Gateaux微分。

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.