关于凸集的适当分离

IF 0.9 4区数学 Q3 MATHEMATICS, APPLIED

Mathematical Methods of Operations Research Pub Date : 2024-06-05 DOI:10.1007/s00186-024-00862-3

Mahmood Mehdiloo

{"title":"关于凸集的适当分离","authors":"Mahmood Mehdiloo","doi":"10.1007/s00186-024-00862-3","DOIUrl":null,"url":null,"abstract":"<p>The aim of this contribution is to propose an alternative but equivalent statement of the proper separation of two closed convex sets in a finite-dimensional Euclidean space. To this aim, we characterize the affine hull of a closed convex set defined by a finite set of equalities and inequalities. Furthermore, we describe algebraically the relative interior of this set by projecting the optimal set of a convex optimization problem onto a subspace of its variables. Then we use this description to develop a system of equalities and inequalities by which the proper separability of the given convex sets is identified. We show that this system is linear in the special case that the given sets are polyhedral.</p>","PeriodicalId":49862,"journal":{"name":"Mathematical Methods of Operations Research","volume":"46 1","pages":""},"PeriodicalIF":0.9000,"publicationDate":"2024-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On proper separation of convex sets\",\"authors\":\"Mahmood Mehdiloo\",\"doi\":\"10.1007/s00186-024-00862-3\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>The aim of this contribution is to propose an alternative but equivalent statement of the proper separation of two closed convex sets in a finite-dimensional Euclidean space. To this aim, we characterize the affine hull of a closed convex set defined by a finite set of equalities and inequalities. Furthermore, we describe algebraically the relative interior of this set by projecting the optimal set of a convex optimization problem onto a subspace of its variables. Then we use this description to develop a system of equalities and inequalities by which the proper separability of the given convex sets is identified. We show that this system is linear in the special case that the given sets are polyhedral.</p>\",\"PeriodicalId\":49862,\"journal\":{\"name\":\"Mathematical Methods of Operations Research\",\"volume\":\"46 1\",\"pages\":\"\"},\"PeriodicalIF\":0.9000,\"publicationDate\":\"2024-06-05\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Mathematical Methods of Operations Research\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1007/s00186-024-00862-3\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS, APPLIED\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematical Methods of Operations Research","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00186-024-00862-3","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}

引用次数: 0

摘要

本论文的目的是就有限维欧几里得空间中两个封闭凸集的适当分离提出另一种等价的说法。为此，我们描述了由有限的等式和不等式集定义的封闭凸集的仿射全形。此外，通过将凸优化问题的最优集投影到其变量子空间上，我们用代数方法描述了该集合的相对内部。然后，我们利用这一描述建立一个等式和不等式系统，通过该系统确定给定凸集的适当可分性。我们证明，在给定集合是多面体的特殊情况下，这个系统是线性的。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

On proper separation of convex sets

查看原文本刊更多论文

On proper separation of convex sets

The aim of this contribution is to propose an alternative but equivalent statement of the proper separation of two closed convex sets in a finite-dimensional Euclidean space. To this aim, we characterize the affine hull of a closed convex set defined by a finite set of equalities and inequalities. Furthermore, we describe algebraically the relative interior of this set by projecting the optimal set of a convex optimization problem onto a subspace of its variables. Then we use this description to develop a system of equalities and inequalities by which the proper separability of the given convex sets is identified. We show that this system is linear in the special case that the given sets are polyhedral.

求助全文

通过发布文献求助，成功后即可免费获取论文全文。去求助

来源期刊

Mathematical Methods of Operations Research 数学-应用数学

CiteScore

1.90

自引率

0.00%

发文量

审稿时长

>12 weeks

期刊介绍： This peer reviewed journal publishes original and high-quality articles on important mathematical and computational aspects of operations research, in particular in the areas of continuous and discrete mathematical optimization, stochastics, and game theory. Theoretically oriented papers are supposed to include explicit motivations of assumptions and results, while application oriented papers need to contain substantial mathematical contributions. Suggestions for algorithms should be accompanied with numerical evidence for their superiority over state-of-the-art methods. Articles must be of interest for a large audience in operations research, written in clear and correct English, and typeset in LaTeX. A special section contains invited tutorial papers on advanced mathematical or computational aspects of operations research, aiming at making such methodologies accessible for a wider audience. All papers are refereed. The emphasis is on originality, quality, and importance.