Eigenvalue programming beyond matrices

IF 16.4 1区 化学 Q1 CHEMISTRY, MULTIDISCIPLINARY
Masaru Ito, Bruno F. Lourenço
{"title":"Eigenvalue programming beyond matrices","authors":"Masaru Ito, Bruno F. Lourenço","doi":"10.1007/s10589-024-00591-7","DOIUrl":null,"url":null,"abstract":"<p>In this paper we analyze and solve eigenvalue programs, which consist of the task of minimizing a function subject to constraints on the “eigenvalues” of the decision variable. Here, by making use of the FTvN systems framework introduced by Gowda, we interpret “eigenvalues” in a broad fashion going beyond the usual eigenvalues of matrices. This allows us to shed new light on classical problems such as inverse eigenvalue problems and also leads to new applications. In particular, after analyzing and developing a simple projected gradient algorithm for general eigenvalue programs, we show that eigenvalue programs can be used to express what we call <i>vanishing quadratic constraints</i>. A vanishing quadratic constraint requires that a given system of convex quadratic inequalities be satisfied and at least a certain number of those inequalities must be tight. As a particular case, this includes the problem of finding a point <i>x</i> in the intersection of <i>m</i> ellipsoids in such a way that <i>x</i> is also in the boundary of at least <span>\\(\\ell \\)</span> of the ellipsoids, for some fixed <span>\\(\\ell &gt; 0\\)</span>. At the end, we also present some numerical experiments.</p>","PeriodicalId":1,"journal":{"name":"Accounts of Chemical Research","volume":null,"pages":null},"PeriodicalIF":16.4000,"publicationDate":"2024-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Accounts of Chemical Research","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s10589-024-00591-7","RegionNum":1,"RegionCategory":"化学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"CHEMISTRY, MULTIDISCIPLINARY","Score":null,"Total":0}
引用次数: 0

Abstract

In this paper we analyze and solve eigenvalue programs, which consist of the task of minimizing a function subject to constraints on the “eigenvalues” of the decision variable. Here, by making use of the FTvN systems framework introduced by Gowda, we interpret “eigenvalues” in a broad fashion going beyond the usual eigenvalues of matrices. This allows us to shed new light on classical problems such as inverse eigenvalue problems and also leads to new applications. In particular, after analyzing and developing a simple projected gradient algorithm for general eigenvalue programs, we show that eigenvalue programs can be used to express what we call vanishing quadratic constraints. A vanishing quadratic constraint requires that a given system of convex quadratic inequalities be satisfied and at least a certain number of those inequalities must be tight. As a particular case, this includes the problem of finding a point x in the intersection of m ellipsoids in such a way that x is also in the boundary of at least \(\ell \) of the ellipsoids, for some fixed \(\ell > 0\). At the end, we also present some numerical experiments.

Abstract Image

超越矩阵的特征值编程
在本文中,我们分析并求解了特征值程序,该程序包括在决策变量 "特征值 "的约束下最小化函数的任务。在这里,通过利用高达提出的 FTvN 系统框架,我们对 "特征值 "进行了广义的解释,超越了通常的矩阵特征值。这使我们能够为逆特征值问题等经典问题带来新的启示,同时也带来了新的应用。特别是,在分析和开发了一般特征值程序的简单投影梯度算法后,我们证明特征值程序可以用来表达我们称之为消失二次约束的问题。消失二次约束要求满足给定的凸二次不等式系统,并且其中至少有一定数量的不等式必须是严密的。作为一种特殊情况,这包括在 m 个椭圆的交点上找到一个点 x,使得 x 在某个固定的 \(\ell > 0\) 的情况下,也在至少 \(\ell \) 个椭圆的边界上。最后,我们还介绍了一些数值实验。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
求助全文
约1分钟内获得全文 求助全文
来源期刊
Accounts of Chemical Research
Accounts of Chemical Research 化学-化学综合
CiteScore
31.40
自引率
1.10%
发文量
312
审稿时长
2 months
期刊介绍: Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
确定
请完成安全验证×
copy
已复制链接
快去分享给好友吧!
我知道了
右上角分享
点击右上角分享
0
联系我们:info@booksci.cn Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。 Copyright © 2023 布克学术 All rights reserved.
京ICP备2023020795号-1
ghs 京公网安备 11010802042870号
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术官方微信