{"title":"Moment-SOS 层次结构：应用和相关主题","authors":"Jean B. Lasserre","doi":"10.1017/s0962492923000053","DOIUrl":null,"url":null,"abstract":"<p>The Moment-SOS hierarchy, first introduced in optimization in 2000, is based on the theory of the <span>S</span>-moment problem and its dual counterpart: polynomials that are positive on <span>S</span>. It turns out that this methodology can also be used to solve problems with positivity constraints ‘<span>f</span>(x) ≥ 0 for all <span><span><span data-mathjax-type=\"texmath\"><span>$\\mathbf{x}\\in S$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240904050110862-0140:S0962492923000053:S0962492923000053_inline1.png\"/></span></span>’ or linear constraints on Borel measures. Such problems can be viewed as specific instances of the <span>generalized moment problem</span> (GMP), whose list of important applications in various domains of science and engineering is almost endless. We describe this methodology in optimization and also in two other applications for illustration. Finally we also introduce the Christoffel function and reveal its links with the Moment-SOS hierarchy and positive polynomials.</p>","PeriodicalId":48863,"journal":{"name":"Acta Numerica","volume":"51 1","pages":""},"PeriodicalIF":16.3000,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"The Moment-SOS hierarchy: Applications and related topics\",\"authors\":\"Jean B. Lasserre\",\"doi\":\"10.1017/s0962492923000053\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>The Moment-SOS hierarchy, first introduced in optimization in 2000, is based on the theory of the <span>S</span>-moment problem and its dual counterpart: polynomials that are positive on <span>S</span>. It turns out that this methodology can also be used to solve problems with positivity constraints ‘<span>f</span>(x) ≥ 0 for all <span><span><span data-mathjax-type=\\\"texmath\\\"><span>$\\\\mathbf{x}\\\\in S$</span></span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240904050110862-0140:S0962492923000053:S0962492923000053_inline1.png\\\"/></span></span>’ or linear constraints on Borel measures. Such problems can be viewed as specific instances of the <span>generalized moment problem</span> (GMP), whose list of important applications in various domains of science and engineering is almost endless. We describe this methodology in optimization and also in two other applications for illustration. Finally we also introduce the Christoffel function and reveal its links with the Moment-SOS hierarchy and positive polynomials.</p>\",\"PeriodicalId\":48863,\"journal\":{\"name\":\"Acta Numerica\",\"volume\":\"51 1\",\"pages\":\"\"},\"PeriodicalIF\":16.3000,\"publicationDate\":\"2024-09-04\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Acta Numerica\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1017/s0962492923000053\",\"RegionNum\":1,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Acta Numerica","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1017/s0962492923000053","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
摘要
2000 年，Moment-SOS 层次结构首次被引入优化领域，它基于 S 时刻问题及其对偶问题的理论：在 S 上为正的多项式。事实证明，这种方法也可以用来解决具有正约束条件 "f(x) ≥ 0 for all $\mathbf{x}\in S$"或伯尔量的线性约束条件的问题。这些问题可以看作是广义矩问题（GMP）的具体实例，而广义矩问题在科学和工程学各个领域的重要应用不胜枚举。我们将在优化和其他两个应用中介绍这种方法，以资说明。最后，我们还介绍了 Christoffel 函数，并揭示了它与矩-SOS 层次和正多项式之间的联系。
The Moment-SOS hierarchy: Applications and related topics
The Moment-SOS hierarchy, first introduced in optimization in 2000, is based on the theory of the S-moment problem and its dual counterpart: polynomials that are positive on S. It turns out that this methodology can also be used to solve problems with positivity constraints ‘f(x) ≥ 0 for all $\mathbf{x}\in S$’ or linear constraints on Borel measures. Such problems can be viewed as specific instances of the generalized moment problem (GMP), whose list of important applications in various domains of science and engineering is almost endless. We describe this methodology in optimization and also in two other applications for illustration. Finally we also introduce the Christoffel function and reveal its links with the Moment-SOS hierarchy and positive polynomials.
期刊介绍：
Acta Numerica stands as the preeminent mathematics journal, ranking highest in both Impact Factor and MCQ metrics. This annual journal features a collection of review articles that showcase survey papers authored by prominent researchers in numerical analysis, scientific computing, and computational mathematics. These papers deliver comprehensive overviews of recent advances, offering state-of-the-art techniques and analyses.
Encompassing the entirety of numerical analysis, the articles are crafted in an accessible style, catering to researchers at all levels and serving as valuable teaching aids for advanced instruction. The broad subject areas covered include computational methods in linear algebra, optimization, ordinary and partial differential equations, approximation theory, stochastic analysis, nonlinear dynamical systems, as well as the application of computational techniques in science and engineering. Acta Numerica also delves into the mathematical theory underpinning numerical methods, making it a versatile and authoritative resource in the field of mathematics.
’ or linear constraints on Borel measures. Such problems can be viewed as specific instances of the generalized moment problem (GMP), whose list of important applications in various domains of science and engineering is almost endless. We describe this methodology in optimization and also in two other applications for illustration. Finally we also introduce the Christoffel function and reveal its links with the Moment-SOS hierarchy and positive polynomials.
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