{"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}
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Abstract
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.
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.
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