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. |
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| 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] |
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| Database: | Engineering Source |
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| 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] |
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| ISSN: | 16153375 |
| DOI: | 10.1007/s10208-025-09691-0 |
