{"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}
引用次数: 0
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.
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.
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