Pub Date : 2025-01-01Epub Date: 2025-03-12DOI: 10.1007/s10013-024-00720-z
Yurii Nesterov
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.
{"title":"Quartic Regularity.","authors":"Yurii Nesterov","doi":"10.1007/s10013-024-00720-z","DOIUrl":"10.1007/s10013-024-00720-z","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":0.8,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12202589/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144530292","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-01-10DOI: 10.1007/s10013-023-00679-3
L. El Fadil, O. Kchit
{"title":"On Index Divisors and Monogenity of Certain Sextic Number Fields Defined by $$x^6+ax^5+b$$","authors":"L. El Fadil, O. Kchit","doi":"10.1007/s10013-023-00679-3","DOIUrl":"https://doi.org/10.1007/s10013-023-00679-3","url":null,"abstract":"","PeriodicalId":45919,"journal":{"name":"Vietnam Journal of Mathematics","volume":"61 15","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139441282","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-01-10DOI: 10.1007/s10013-023-00678-4
Oanh Thi Dang
{"title":"An Improved Adaptive Meshless Refinement for the RBF-FD Method for 2D Elliptic Equations","authors":"Oanh Thi Dang","doi":"10.1007/s10013-023-00678-4","DOIUrl":"https://doi.org/10.1007/s10013-023-00678-4","url":null,"abstract":"","PeriodicalId":45919,"journal":{"name":"Vietnam Journal of Mathematics","volume":"62 13","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139441263","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-01-09DOI: 10.1007/s10013-023-00669-5
M. Dindoš, Erik Sätterqvist, Martin Ulmer
{"title":"Correction: Perturbation Theory for Second Order Elliptic Operators with BMO Antisymmetric Part","authors":"M. Dindoš, Erik Sätterqvist, Martin Ulmer","doi":"10.1007/s10013-023-00669-5","DOIUrl":"https://doi.org/10.1007/s10013-023-00669-5","url":null,"abstract":"","PeriodicalId":45919,"journal":{"name":"Vietnam Journal of Mathematics","volume":"24 8","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-01-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139441346","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-01-04DOI: 10.1007/s10013-023-00676-6
S. Gala, M. Théra
{"title":"Uniform Regularity Estimates for Three Dimensional Nonhomogeneous Incompressible Heat Conducting Navier–Stokes Flows","authors":"S. Gala, M. Théra","doi":"10.1007/s10013-023-00676-6","DOIUrl":"https://doi.org/10.1007/s10013-023-00676-6","url":null,"abstract":"","PeriodicalId":45919,"journal":{"name":"Vietnam Journal of Mathematics","volume":"3 4","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-01-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139384388","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-01-04DOI: 10.1007/s10013-023-00675-7
Noriyuki Abe, Florian Herzig
{"title":"On the Irreducibility of p-Adic Banach Principal Series of p-Adic $$textrm{GL}_3$$","authors":"Noriyuki Abe, Florian Herzig","doi":"10.1007/s10013-023-00675-7","DOIUrl":"https://doi.org/10.1007/s10013-023-00675-7","url":null,"abstract":"","PeriodicalId":45919,"journal":{"name":"Vietnam Journal of Mathematics","volume":"56 40","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-01-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139384733","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-12-06DOI: 10.1007/s10013-023-00664-w
Felipe Linares, Argenis J. Mendez, D. Pilod
{"title":"Well-Posedness for the Extended Schrödinger–Benjamin–Ono System","authors":"Felipe Linares, Argenis J. Mendez, D. Pilod","doi":"10.1007/s10013-023-00664-w","DOIUrl":"https://doi.org/10.1007/s10013-023-00664-w","url":null,"abstract":"","PeriodicalId":45919,"journal":{"name":"Vietnam Journal of Mathematics","volume":"67 42","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138594846","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-12-02DOI: 10.1007/s10013-023-00667-7
Sami Baraket, Rima Chetouane, Rached Jaidane
{"title":"Signed and Sign-Changing Solutions for a Kirchhoff-Type Problem Involving the Weighted N-Laplacian with Critical Double Exponential Growth","authors":"Sami Baraket, Rima Chetouane, Rached Jaidane","doi":"10.1007/s10013-023-00667-7","DOIUrl":"https://doi.org/10.1007/s10013-023-00667-7","url":null,"abstract":"","PeriodicalId":45919,"journal":{"name":"Vietnam Journal of Mathematics","volume":"68 11","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138606473","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-12-02DOI: 10.1007/s10013-023-00662-y
S. Soltani Renani, Z. Yari
{"title":"Characterization of Weakly Compact Multipliers on Left $$phi $$-Amenable Banach Algebras","authors":"S. Soltani Renani, Z. Yari","doi":"10.1007/s10013-023-00662-y","DOIUrl":"https://doi.org/10.1007/s10013-023-00662-y","url":null,"abstract":"","PeriodicalId":45919,"journal":{"name":"Vietnam Journal of Mathematics","volume":"118 46","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138607240","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-29DOI: 10.1007/s10013-023-00661-z
Alexandru D. Ionescu, Sameer Iyer, Hao Jia
{"title":"On the Stability of Shear Flows in Bounded Channels, II: Non-monotonic Shear Flows","authors":"Alexandru D. Ionescu, Sameer Iyer, Hao Jia","doi":"10.1007/s10013-023-00661-z","DOIUrl":"https://doi.org/10.1007/s10013-023-00661-z","url":null,"abstract":"","PeriodicalId":45919,"journal":{"name":"Vietnam Journal of Mathematics","volume":"26 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139213509","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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