Active Manifolds, Stratifications, and Convergence to Local Minima in Nonsmooth Optimization.
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| Title: | Active Manifolds, Stratifications, and Convergence to Local Minima in Nonsmooth Optimization. |
|---|---|
| Authors: | Davis, Damek1 (AUTHOR), Drusvyatskiy, Dmitriy2 (AUTHOR) ddrusv@uw.edu, Jiang, Liwei1 (AUTHOR) |
| Source: | Foundations of Computational Mathematics. Apr2026, Vol. 26 Issue 2, p779-861. 83p. |
| Subjects: | Nonsmooth optimization, Subgradient methods, Stochastic processes, Lipschitz spaces, Submanifolds, Critical point theory |
| Abstract: | In this work, we develop new regularity conditions in nonsmooth analysis that parallel the stratification conditions of Whitney, Kuo, and Verdier. They quantify how subgradients interact with a certain "active manifold" that captures the nonsmooth activity of the function. Based on these new conditions, we show that several subgradient-based methods converge only to local minimizers when applied to generic Lipschitz and subdifferentially regular functions that are definable in an o-minimal structure. At a high level, our argument is appealingly transparent: we interpret the nonsmooth dynamics as an approximate Riemannian gradient method on the active manifold. As a by-product, we extend the stochastic processes techniques of Pemantle. [ABSTRACT FROM AUTHOR] |
| Copyright of Foundations of Computational Mathematics 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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| Header | DbId: egs DbLabel: Engineering Source An: 193283912 AccessLevel: 6 PubType: Academic Journal PubTypeId: academicJournal PreciseRelevancyScore: 0 |
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| Items | – Name: Title Label: Title Group: Ti Data: Active Manifolds, Stratifications, and Convergence to Local Minima in Nonsmooth Optimization. – Name: Author Label: Authors Group: Au Data: <searchLink fieldCode="AR" term="%22Davis%2C+Damek%22">Davis, Damek</searchLink><relatesTo>1</relatesTo> (AUTHOR)<br /><searchLink fieldCode="AR" term="%22Drusvyatskiy%2C+Dmitriy%22">Drusvyatskiy, Dmitriy</searchLink><relatesTo>2</relatesTo> (AUTHOR)<i> ddrusv@uw.edu</i><br /><searchLink fieldCode="AR" term="%22Jiang%2C+Liwei%22">Jiang, Liwei</searchLink><relatesTo>1</relatesTo> (AUTHOR) – Name: TitleSource Label: Source Group: Src Data: <searchLink fieldCode="JN" term="%22Foundations+of+Computational+Mathematics%22">Foundations of Computational Mathematics</searchLink>. Apr2026, Vol. 26 Issue 2, p779-861. 83p. – Name: Subject Label: Subjects Group: Su Data: <searchLink fieldCode="DE" term="%22Nonsmooth+optimization%22">Nonsmooth optimization</searchLink><br /><searchLink fieldCode="DE" term="%22Subgradient+methods%22">Subgradient methods</searchLink><br /><searchLink fieldCode="DE" term="%22Stochastic+processes%22">Stochastic processes</searchLink><br /><searchLink fieldCode="DE" term="%22Lipschitz+spaces%22">Lipschitz spaces</searchLink><br /><searchLink fieldCode="DE" term="%22Submanifolds%22">Submanifolds</searchLink><br /><searchLink fieldCode="DE" term="%22Critical+point+theory%22">Critical point theory</searchLink> – Name: Abstract Label: Abstract Group: Ab Data: In this work, we develop new regularity conditions in nonsmooth analysis that parallel the stratification conditions of Whitney, Kuo, and Verdier. They quantify how subgradients interact with a certain "active manifold" that captures the nonsmooth activity of the function. Based on these new conditions, we show that several subgradient-based methods converge only to local minimizers when applied to generic Lipschitz and subdifferentially regular functions that are definable in an o-minimal structure. At a high level, our argument is appealingly transparent: we interpret the nonsmooth dynamics as an approximate Riemannian gradient method on the active manifold. As a by-product, we extend the stochastic processes techniques of Pemantle. [ABSTRACT FROM AUTHOR] – Name: AbstractSuppliedCopyright Label: Group: Ab Data: <i>Copyright of Foundations of Computational Mathematics 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/s10208-025-09691-0 Languages: – Code: eng Text: English PhysicalDescription: Pagination: PageCount: 83 StartPage: 779 Subjects: – SubjectFull: Nonsmooth optimization Type: general – SubjectFull: Subgradient methods Type: general – SubjectFull: Stochastic processes Type: general – SubjectFull: Lipschitz spaces Type: general – SubjectFull: Submanifolds Type: general – SubjectFull: Critical point theory Type: general Titles: – TitleFull: Active Manifolds, Stratifications, and Convergence to Local Minima in Nonsmooth Optimization. Type: main BibRelationships: HasContributorRelationships: – PersonEntity: Name: NameFull: Davis, Damek – PersonEntity: Name: NameFull: Drusvyatskiy, Dmitriy – PersonEntity: Name: NameFull: Jiang, Liwei IsPartOfRelationships: – BibEntity: Dates: – D: 01 M: 04 Text: Apr2026 Type: published Y: 2026 Identifiers: – Type: issn-print Value: 16153375 Numbering: – Type: volume Value: 26 – Type: issue Value: 2 Titles: – TitleFull: Foundations of Computational Mathematics Type: main |
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