论文标题
无拓扑的代数核心和凸积分
Algebraic Core and Convex Calculus without Topology
论文作者
论文摘要
在本文中，我们研究了在没有任何拓扑结构的一般矢量空间中的凸组集合的代数核心的概念，然后将其应用于凸分析和优化问题。得出Hahn-Banach定理与矢量空间中凸的分离定理的简单版本之间的等效性，使我们能够开发出一种几何方法，以用于凸面集的通用差分计算，设置值映射，具有延长的价值功能，并具有与符号的相关条件，以对对象进行构图。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佳值函数的细分。这种类型的功能在凸优化及其应用的许多方面都起着至关重要的作用。
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.
