Homogeneous second-order descent framework: a fast alternative to Newton-type methods.
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| Title: | Homogeneous second-order descent framework: a fast alternative to Newton-type methods. |
|---|---|
| Authors: | He, Chang1 (AUTHOR) ischanghe@gmail.com, Jiang, Yuntian1 (AUTHOR) yuntianjiang07@163.sufe.edu.cn, Zhang, Chuwen1 (AUTHOR) chuwzhang@gmail.com, Ge, Dongdong2 (AUTHOR) ddge@sjtu.edu.cn, Jiang, Bo1 (AUTHOR) isyebojiang@gmail.com, Ye, Yinyu3 (AUTHOR) yyye@stanford.edu |
| Source: | Mathematical Programming. Jan2026, Vol. 215 Issue 1/2, p575-636. 62p. |
| Subjects: | Nonconvex programming, Mathematical optimization, Subgradient methods, Convex programming |
| Abstract: | This paper proposes a homogeneous second-order descent framework (HSODF) for nonconvex and convex optimization based on the generalized homogeneous model (GHM). In comparison to the Newton steps, the GHM can be solved by extremal symmetric eigenvalue procedures and thus grant an advantage in ill-conditioned problems. Moreover, GHM extends the ordinary homogeneous model (Zhang et al. A homogenous second-order descent method for nonconvex optimization, 2022. arXiv:2211.08212 [math]) to allow adaptiveness in the construction of the aggregated matrix. Consequently, HSODF is able to recover some well-known second-order methods, such as trust-region methods and gradient regularized methods, while maintaining comparable iteration complexity bounds. We also study two specific realizations of HSODF. One is adaptive HSODM, which has a parameter-free O (ϵ - 3 / 2) global complexity bound for nonconvex second-order Lipschitz continuous objective functions. The other is homotopy HSODM, which is proven to have a global linear rate of convergence without strong convexity. The efficiency of our approach to high-dimensional and ill-conditioned problems is justified by some preliminary numerical results. [ABSTRACT FROM AUTHOR] |
| Copyright of Mathematical Programming is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.) | |
| Database: | Engineering Source |
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| Items | – Name: Title Label: Title Group: Ti Data: Homogeneous second-order descent framework: a fast alternative to Newton-type methods. – Name: Author Label: Authors Group: Au Data: <searchLink fieldCode="AR" term="%22He%2C+Chang%22">He, Chang</searchLink><relatesTo>1</relatesTo> (AUTHOR)<i> ischanghe@gmail.com</i><br /><searchLink fieldCode="AR" term="%22Jiang%2C+Yuntian%22">Jiang, Yuntian</searchLink><relatesTo>1</relatesTo> (AUTHOR)<i> yuntianjiang07@163.sufe.edu.cn</i><br /><searchLink fieldCode="AR" term="%22Zhang%2C+Chuwen%22">Zhang, Chuwen</searchLink><relatesTo>1</relatesTo> (AUTHOR)<i> chuwzhang@gmail.com</i><br /><searchLink fieldCode="AR" term="%22Ge%2C+Dongdong%22">Ge, Dongdong</searchLink><relatesTo>2</relatesTo> (AUTHOR)<i> ddge@sjtu.edu.cn</i><br /><searchLink fieldCode="AR" term="%22Jiang%2C+Bo%22">Jiang, Bo</searchLink><relatesTo>1</relatesTo> (AUTHOR)<i> isyebojiang@gmail.com</i><br /><searchLink fieldCode="AR" term="%22Ye%2C+Yinyu%22">Ye, Yinyu</searchLink><relatesTo>3</relatesTo> (AUTHOR)<i> yyye@stanford.edu</i> – Name: TitleSource Label: Source Group: Src Data: <searchLink fieldCode="JN" term="%22Mathematical+Programming%22">Mathematical Programming</searchLink>. Jan2026, Vol. 215 Issue 1/2, p575-636. 62p. – Name: Subject Label: Subjects Group: Su Data: <searchLink fieldCode="DE" term="%22Nonconvex+programming%22">Nonconvex programming</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematical+optimization%22">Mathematical optimization</searchLink><br /><searchLink fieldCode="DE" term="%22Subgradient+methods%22">Subgradient methods</searchLink><br /><searchLink fieldCode="DE" term="%22Convex+programming%22">Convex programming</searchLink> – Name: Abstract Label: Abstract Group: Ab Data: This paper proposes a homogeneous second-order descent framework (HSODF) for nonconvex and convex optimization based on the generalized homogeneous model (GHM). In comparison to the Newton steps, the GHM can be solved by extremal symmetric eigenvalue procedures and thus grant an advantage in ill-conditioned problems. Moreover, GHM extends the ordinary homogeneous model (Zhang et al. A homogenous second-order descent method for nonconvex optimization, 2022. arXiv:2211.08212 [math]) to allow adaptiveness in the construction of the aggregated matrix. Consequently, HSODF is able to recover some well-known second-order methods, such as trust-region methods and gradient regularized methods, while maintaining comparable iteration complexity bounds. We also study two specific realizations of HSODF. One is adaptive HSODM, which has a parameter-free O (ϵ - 3 / 2) global complexity bound for nonconvex second-order Lipschitz continuous objective functions. The other is homotopy HSODM, which is proven to have a global linear rate of convergence without strong convexity. The efficiency of our approach to high-dimensional and ill-conditioned problems is justified by some preliminary numerical results. [ABSTRACT FROM AUTHOR] – Name: AbstractSuppliedCopyright Label: Group: Ab Data: <i>Copyright of Mathematical Programming is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.) |
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| RecordInfo | BibRecord: BibEntity: Identifiers: – Type: doi Value: 10.1007/s10107-025-02230-3 Languages: – Code: eng Text: English PhysicalDescription: Pagination: PageCount: 62 StartPage: 575 Subjects: – SubjectFull: Nonconvex programming Type: general – SubjectFull: Mathematical optimization Type: general – SubjectFull: Subgradient methods Type: general – SubjectFull: Convex programming Type: general Titles: – TitleFull: Homogeneous second-order descent framework: a fast alternative to Newton-type methods. Type: main BibRelationships: HasContributorRelationships: – PersonEntity: Name: NameFull: He, Chang – PersonEntity: Name: NameFull: Jiang, Yuntian – PersonEntity: Name: NameFull: Zhang, Chuwen – PersonEntity: Name: NameFull: Ge, Dongdong – PersonEntity: Name: NameFull: Jiang, Bo – PersonEntity: Name: NameFull: Ye, Yinyu IsPartOfRelationships: – BibEntity: Dates: – D: 01 M: 01 Text: Jan2026 Type: published Y: 2026 Identifiers: – Type: issn-print Value: 00255610 Numbering: – Type: volume Value: 215 – Type: issue Value: 1/2 Titles: – TitleFull: Mathematical Programming Type: main |
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