ABSTRACT

In this paper we study the concept of algebraic core for convex sets in general vector spaces without any topological structure and then present its applications to problems of convex analysis and optimization. Deriving the equivalence between the Hahn-Banach theorem and and a simple version of the separation theorem of convex sets in vector spaces allows us to develop a geometric approach to generalized differential calculus for convex sets, set-valued mappings, and extended-real-valued functions with qualification conditions formulated in terms of algebraic cores for such objects. We also obtain a precise formula for computing the subdifferential of optimal value functions associated with convex problems of parametric optimization in vector spaces. Functions of this type play a crucial role in many aspects of convex optimization and its applications.

摘要

在本文中，我们研究了没有任何拓扑结构的一般向量空间中凸集的代数核的概念，然后介绍了它在凸分析和优化问题中的应用。推导 Hahn-Banach 定理和向量空间中凸集分离定理的简单版本之间的等价性，使我们能够开发一种几何方法来计算凸集、集值映射和扩展实值的广义微分具有根据此类对象的代数核心制定的限定条件的函数。我们还获得了一个精确的公式，用于计算与向量空间中参数优化的凸问题相关的最优值函数的次微分。