关于带有指示变量的二次约束的说明：凸壳描述与透视松弛

IF 0.8 4区管理学 Q4 OPERATIONS RESEARCH & MANAGEMENT SCIENCE

Operations Research Letters Pub Date : 2024-01-01 DOI:10.1016/j.orl.2023.107059

Andrés Gómez , Weijun Xie

{"title":"关于带有指示变量的二次约束的说明：凸壳描述与透视松弛","authors":"Andrés Gómez , Weijun Xie","doi":"10.1016/j.orl.2023.107059","DOIUrl":null,"url":null,"abstract":"<div><p>In this paper, we study the mixed-integer nonlinear set given by a separable quadratic constraint<span> on continuous variables, where each continuous variable is controlled by an additional indicator. This set occurs pervasively in optimization problems with uncertainty and in machine learning. We show that optimization over this set is NP-hard. Despite this negative result, we discover links between the convex hull of the set under study, and a family of polyhedral sets studied in the literature. Moreover, we show that although perspective relaxation in the literature for this set fails to match the structure of its convex hull, it is guaranteed to be a close approximation.</span></p></div>","PeriodicalId":54682,"journal":{"name":"Operations Research Letters","volume":"52 ","pages":"Article 107059"},"PeriodicalIF":0.8000,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"A note on quadratic constraints with indicator variables: Convex hull description and perspective relaxation\",\"authors\":\"Andrés Gómez , Weijun Xie\",\"doi\":\"10.1016/j.orl.2023.107059\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>In this paper, we study the mixed-integer nonlinear set given by a separable quadratic constraint<span> on continuous variables, where each continuous variable is controlled by an additional indicator. This set occurs pervasively in optimization problems with uncertainty and in machine learning. We show that optimization over this set is NP-hard. Despite this negative result, we discover links between the convex hull of the set under study, and a family of polyhedral sets studied in the literature. Moreover, we show that although perspective relaxation in the literature for this set fails to match the structure of its convex hull, it is guaranteed to be a close approximation.</span></p></div>\",\"PeriodicalId\":54682,\"journal\":{\"name\":\"Operations Research Letters\",\"volume\":\"52 \",\"pages\":\"Article 107059\"},\"PeriodicalIF\":0.8000,\"publicationDate\":\"2024-01-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Operations Research Letters\",\"FirstCategoryId\":\"91\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0167637723002006\",\"RegionNum\":4,\"RegionCategory\":\"管理学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"OPERATIONS RESEARCH & MANAGEMENT SCIENCE\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Operations Research Letters","FirstCategoryId":"91","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0167637723002006","RegionNum":4,"RegionCategory":"管理学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"OPERATIONS RESEARCH & MANAGEMENT SCIENCE","Score":null,"Total":0}

引用次数: 0

摘要

在本文中，我们将研究由连续变量的可分离二次约束给出的混合整数非线性集合，其中每个连续变量都由一个附加指标控制。这个集合普遍存在于具有不确定性的优化问题和机器学习中。我们的研究表明，对这个集合进行优化是 NP-困难的。尽管结果是否定的，但我们发现了所研究集合的凸壳与文献中研究的多面体集合家族之间的联系。此外，我们还证明，虽然文献中针对该集合的透视松弛与凸壳的结构不匹配，但可以保证是近似的。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

查看原文本刊更多论文

A note on quadratic constraints with indicator variables: Convex hull description and perspective relaxation

In this paper, we study the mixed-integer nonlinear set given by a separable quadratic constraint on continuous variables, where each continuous variable is controlled by an additional indicator. This set occurs pervasively in optimization problems with uncertainty and in machine learning. We show that optimization over this set is NP-hard. Despite this negative result, we discover links between the convex hull of the set under study, and a family of polyhedral sets studied in the literature. Moreover, we show that although perspective relaxation in the literature for this set fails to match the structure of its convex hull, it is guaranteed to be a close approximation.

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

Operations Research Letters 管理科学-运筹学与管理科学

CiteScore

2.10

自引率

9.10%

发文量

111

审稿时长

83 days

期刊介绍： Operations Research Letters is committed to the rapid review and fast publication of short articles on all aspects of operations research and analytics. Apart from a limitation to eight journal pages, quality, originality, relevance and clarity are the only criteria for selecting the papers to be published. ORL covers the broad field of optimization, stochastic models and game theory. Specific areas of interest include networks, routing, location, queueing, scheduling, inventory, reliability, and financial engineering. We wish to explore interfaces with other fields such as life sciences and health care, artificial intelligence and machine learning, energy distribution, and computational social sciences and humanities. Our traditional strength is in methodology, including theory, modelling, algorithms and computational studies. We also welcome novel applications and concise literature reviews.