How does the contraction property fail for convex functions on normed spaces?
Fuente:
arXiv
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| Autore principale: | |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866908455055392768 |
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| author | Ohta, Shin-ichi |
| author_facet | Ohta, Shin-ichi |
| contents | On Euclidean and Hilbert spaces, Riemannian manifolds, and CAT$(0)$-spaces, gradient flows of convex functions are known to satisfy the contraction property, which plays a fundamental role in optimization theory and possesses fruitful analytic and geometric applications. On (non-inner product) normed spaces, however, gradient flows of convex functions do not satisfy the contraction property. We give a detailed proof of this characterization of inner products, and discuss a possible form of a weaker contraction property on normed spaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_15152 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | How does the contraction property fail for convex functions on normed spaces? Ohta, Shin-ichi Optimization and Control Metric Geometry On Euclidean and Hilbert spaces, Riemannian manifolds, and CAT$(0)$-spaces, gradient flows of convex functions are known to satisfy the contraction property, which plays a fundamental role in optimization theory and possesses fruitful analytic and geometric applications. On (non-inner product) normed spaces, however, gradient flows of convex functions do not satisfy the contraction property. We give a detailed proof of this characterization of inner products, and discuss a possible form of a weaker contraction property on normed spaces. |
| title | How does the contraction property fail for convex functions on normed spaces? |
| topic | Optimization and Control Metric Geometry |
| url | https://arxiv.org/abs/2311.15152 |