论应用于凸函数最小化的牛顿法线性收敛率的黑森非奇异性的冗余性

IF 0.7 4区数学 Q3 MATHEMATICS, APPLIED

Computational Mathematics and Mathematical Physics Pub Date : 2024-06-07 DOI:10.1134/s0965542524700040

Yu. G. Evtushenko, A. A. Tret’yakov

{"title":"论应用于凸函数最小化的牛顿法线性收敛率的黑森非奇异性的冗余性","authors":"Yu. G. Evtushenko, A. A. Tret’yakov","doi":"10.1134/s0965542524700040","DOIUrl":null,"url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>A new property of convex functions that makes it possible to achieve the linear rate of convergence of the Newton method during the minimization process is established. Namely, it is proved that, even in the case of singularity of the Hessian at the solution, the Newtonian system is solvable in the vicinity of the minimizer; i.e., the gradient of the objective function belongs to the image of the matrix of second derivatives and, therefore, analogs of the Newton method may be used.</p>","PeriodicalId":55230,"journal":{"name":"Computational Mathematics and Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":0.7000,"publicationDate":"2024-06-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On the Redundancy of Hessian Nonsingularity for Linear Convergence Rate of the Newton Method Applied to the Minimization of Convex Functions\",\"authors\":\"Yu. G. Evtushenko, A. A. Tret’yakov\",\"doi\":\"10.1134/s0965542524700040\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<h3 data-test=\\\"abstract-sub-heading\\\">Abstract</h3><p>A new property of convex functions that makes it possible to achieve the linear rate of convergence of the Newton method during the minimization process is established. Namely, it is proved that, even in the case of singularity of the Hessian at the solution, the Newtonian system is solvable in the vicinity of the minimizer; i.e., the gradient of the objective function belongs to the image of the matrix of second derivatives and, therefore, analogs of the Newton method may be used.</p>\",\"PeriodicalId\":55230,\"journal\":{\"name\":\"Computational Mathematics and Mathematical Physics\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.7000,\"publicationDate\":\"2024-06-07\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Computational Mathematics and Mathematical Physics\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1134/s0965542524700040\",\"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":"Computational Mathematics and Mathematical Physics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1134/s0965542524700040","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

摘要建立了凸函数的一个新特性，它使牛顿方法在最小化过程中实现线性收敛率成为可能。即，证明了即使在解的 Hessian 存在奇异性的情况下，牛顿系统仍可在最小化附近求解；也就是说，目标函数的梯度属于二阶导数矩阵的图像，因此可以使用牛顿方法的类似方法。

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

On the Redundancy of Hessian Nonsingularity for Linear Convergence Rate of the Newton Method Applied to the Minimization of Convex Functions

Abstract

A new property of convex functions that makes it possible to achieve the linear rate of convergence of the Newton method during the minimization process is established. Namely, it is proved that, even in the case of singularity of the Hessian at the solution, the Newtonian system is solvable in the vicinity of the minimizer; i.e., the gradient of the objective function belongs to the image of the matrix of second derivatives and, therefore, analogs of the Newton method may be used.

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

Computational Mathematics and Mathematical Physics MATHEMATICS, APPLIED-PHYSICS, MATHEMATICAL

CiteScore

1.50

自引率

14.30%

发文量

125

审稿时长

4-8 weeks

期刊介绍： Computational Mathematics and Mathematical Physics is a monthly journal published in collaboration with the Russian Academy of Sciences. The journal includes reviews and original papers on computational mathematics, computational methods of mathematical physics, informatics, and other mathematical sciences. The journal welcomes reviews and original articles from all countries in the English or Russian language.