Existence of extreme points of compact convex sets in asymmetric cone normed spaces

One of the most popular applications in real-world problems from the past into the present is optimization. It is well-known that the concept of extreme points plays a vital role in optimization, and the existence of extreme points of compact convex subsets of a locally convex (Hausdorff) vector space can be obtained from the KreinMilman theorem. As the usefulness of this application, many researchers follow this idea to investigate the existence of extreme points in compact convex subsets of various abstract spaces. In 2016, the research in this direction in asymmetric normed spaces was proved in Jonard-Pérez and Sánchez-Pérez (2016). Another important concept parallel to the concept of extreme points in optimization is the concept of cones. Most recently, the idea of cones is used to extend asymmetric normed spaces to asymmetric cone normed spaces, and its topological properties are studied. Surprisingly, nobody considered the existence of extreme points of compact convex subsets of asymmetric cone normed spaces. Our goal in this research is to fulfill this direction. Hence, the sufficient condition for the existence of extreme points of nonempty compact convex subsets of asymmetric cone normed spaces is invented. An example to illustrate the main result presented herein is given.

从古至今，优化便是现实世界各类问题中最受关注的应用方向之一。众所周知，极点（extreme points）的概念在优化领域中扮演着至关重要的角色，而局部凸（豪斯多夫）向量空间的紧凸子集的极点存在性可由克赖恩-米尔曼定理（Krein-Milman theorem）推导得出。鉴于该理论的应用价值，诸多研究者沿此思路，对各类抽象空间的紧凸子集的极点存在性展开了研究。2016年，Jonard-Pérez与Sánchez-Pérez（2016）针对非对称赋范空间（asymmetric normed spaces）中的该方向开展了相关研究并给出了证明。与优化领域中的极点概念并行的另一重要概念是锥（cones）。近期，研究者借助锥的概念将非对称赋范空间推广至非对称锥赋范空间（asymmetric cone normed spaces），并对其拓扑性质展开了研究。令人意外的是，目前尚无研究者对非对称锥赋范空间的紧凸子集的极点存在性展开探讨。本研究的目标正是填补这一研究空白。据此，本文首次建立了非对称锥赋范空间中非空紧凸子集的极点存在性充分条件。文末还给出了一个示例，用以说明本文所得出的主要结论。