Analyzing the speed of convergence in nonsmooth optimization via the Goldstein subdifferential with application to descent methods
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913892997791744 |
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| author | Gebken, Bennet |
| author_facet | Gebken, Bennet |
| contents | The Goldstein $\varepsilon$-subdifferential is a relaxed version of the Clarke subdifferential which has recently appeared in several algorithms for nonsmooth optimization. With it comes the notion of $(\varepsilon,δ)$-critical points, which are points in which the element with the smallest norm in the $\varepsilon$-subdifferential has norm at most $δ$. To obtain points that are critical in the classical sense, $\varepsilon$ and $δ$ must vanish. In this article, we analyze at which speed the distance of $(\varepsilon,δ)$-critical points to the minimum vanishes with respect to $\varepsilon$ and $δ$. Afterwards, we apply our results to gradient sampling methods and perform numerical experiments. Throughout the article, we put a special emphasis on supporting the theoretical results with simple examples that visualize them. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_01382 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Analyzing the speed of convergence in nonsmooth optimization via the Goldstein subdifferential with application to descent methods Gebken, Bennet Optimization and Control 90C30, 90C56, 49J52 The Goldstein $\varepsilon$-subdifferential is a relaxed version of the Clarke subdifferential which has recently appeared in several algorithms for nonsmooth optimization. With it comes the notion of $(\varepsilon,δ)$-critical points, which are points in which the element with the smallest norm in the $\varepsilon$-subdifferential has norm at most $δ$. To obtain points that are critical in the classical sense, $\varepsilon$ and $δ$ must vanish. In this article, we analyze at which speed the distance of $(\varepsilon,δ)$-critical points to the minimum vanishes with respect to $\varepsilon$ and $δ$. Afterwards, we apply our results to gradient sampling methods and perform numerical experiments. Throughout the article, we put a special emphasis on supporting the theoretical results with simple examples that visualize them. |
| title | Analyzing the speed of convergence in nonsmooth optimization via the Goldstein subdifferential with application to descent methods |
| topic | Optimization and Control 90C30, 90C56, 49J52 |
| url | https://arxiv.org/abs/2410.01382 |