{"title":"四次规律性。","authors":"Yurii Nesterov","doi":"10.1007/s10013-024-00720-z","DOIUrl":null,"url":null,"abstract":"<p><p>In this paper, we propose new linearly convergent second-order methods for minimizing convex quartic polynomials. This framework is applied for designing optimization schemes, which can solve general convex problems satisfying a new condition of quartic regularity. It assumes positive definiteness and boundedness of the fourth derivative of the objective function. For such problems, an appropriate quartic regularization of Damped Newton Method has global linear rate of convergence. We discuss several important consequences of this result. In particular, it can be used for constructing new second-order methods in the framework of high-order proximal-point schemes (Nesterov, Math. Program. <b>197</b>, 1-26, 2023 and Nesterov, SIAM J. Optim. <b>31</b>, 2807-2828, 2021). These methods have convergence rate <math> <mrow><mover><mi>O</mi> <mo>~</mo></mover> <mrow><mo>(</mo> <msup><mi>k</mi> <mrow><mo>-</mo> <mi>p</mi></mrow> </msup> <mo>)</mo></mrow> </mrow> </math> , where <i>k</i> is the iteration counter, <i>p</i> is equal to 3, 4, or 5, and tilde indicates the presence of logarithmic factors in the complexity bounds for the auxiliary problems, which are solved at each iteration of the schemes.</p>","PeriodicalId":45919,"journal":{"name":"Vietnam Journal of Mathematics","volume":"53 3","pages":"553-575"},"PeriodicalIF":0.7000,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12202589/pdf/","citationCount":"0","resultStr":"{\"title\":\"Quartic Regularity.\",\"authors\":\"Yurii Nesterov\",\"doi\":\"10.1007/s10013-024-00720-z\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p><p>In this paper, we propose new linearly convergent second-order methods for minimizing convex quartic polynomials. This framework is applied for designing optimization schemes, which can solve general convex problems satisfying a new condition of quartic regularity. It assumes positive definiteness and boundedness of the fourth derivative of the objective function. For such problems, an appropriate quartic regularization of Damped Newton Method has global linear rate of convergence. We discuss several important consequences of this result. In particular, it can be used for constructing new second-order methods in the framework of high-order proximal-point schemes (Nesterov, Math. Program. <b>197</b>, 1-26, 2023 and Nesterov, SIAM J. Optim. <b>31</b>, 2807-2828, 2021). These methods have convergence rate <math> <mrow><mover><mi>O</mi> <mo>~</mo></mover> <mrow><mo>(</mo> <msup><mi>k</mi> <mrow><mo>-</mo> <mi>p</mi></mrow> </msup> <mo>)</mo></mrow> </mrow> </math> , where <i>k</i> is the iteration counter, <i>p</i> is equal to 3, 4, or 5, and tilde indicates the presence of logarithmic factors in the complexity bounds for the auxiliary problems, which are solved at each iteration of the schemes.</p>\",\"PeriodicalId\":45919,\"journal\":{\"name\":\"Vietnam Journal of Mathematics\",\"volume\":\"53 3\",\"pages\":\"553-575\"},\"PeriodicalIF\":0.7000,\"publicationDate\":\"2025-01-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12202589/pdf/\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Vietnam Journal of Mathematics\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1007/s10013-024-00720-z\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"2025/3/12 0:00:00\",\"PubModel\":\"Epub\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Vietnam Journal of Mathematics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1007/s10013-024-00720-z","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"2025/3/12 0:00:00","PubModel":"Epub","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
In this paper, we propose new linearly convergent second-order methods for minimizing convex quartic polynomials. This framework is applied for designing optimization schemes, which can solve general convex problems satisfying a new condition of quartic regularity. It assumes positive definiteness and boundedness of the fourth derivative of the objective function. For such problems, an appropriate quartic regularization of Damped Newton Method has global linear rate of convergence. We discuss several important consequences of this result. In particular, it can be used for constructing new second-order methods in the framework of high-order proximal-point schemes (Nesterov, Math. Program. 197, 1-26, 2023 and Nesterov, SIAM J. Optim. 31, 2807-2828, 2021). These methods have convergence rate , where k is the iteration counter, p is equal to 3, 4, or 5, and tilde indicates the presence of logarithmic factors in the complexity bounds for the auxiliary problems, which are solved at each iteration of the schemes.
期刊介绍:
Vietnam Journal of Mathematics was originally founded in 1973 by the Vietnam Academy of Science and Technology and the Vietnam Mathematical Society. Published by Springer from 1997 to 2005 and since 2013, this quarterly journal is open to contributions from researchers from all over the world, where all submitted articles are peer-reviewed by experts worldwide. It aims to publish high-quality original research papers and review articles in all active areas of pure and applied mathematics.