On strict proto-differentiability of set-valued mappings
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866913908617379840 |
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| author | Gfrerer, Helmut |
| author_facet | Gfrerer, Helmut |
| contents | We will show that a multifunction is strictly proto-differentiable at a point of its graph if and only if it is graphically strictly differentiable, i.e., the graph of the multifunction locally coincides, up to a change of coordinates, with the graph of a single-valued mapping, which is strictly differentiable at the transformed reference point. This result allows point-based characterizations of strict proto-differentiability in terms of various generalized derivatives. Further we will prove that under strict proto-differentiability the properties of strong metric regularity, metric regularity and strong metric subregularity are equivalent. Finally, under strict proto-differentiability of the subgradient mapping, we provide a novel second-order relation between function values and subgradients for prox-regular functions which constitutes a nonsmooth extension of the trapezoidal rule of numerical integration. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_01346 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On strict proto-differentiability of set-valued mappings Gfrerer, Helmut Optimization and Control 49J53, 90C31 We will show that a multifunction is strictly proto-differentiable at a point of its graph if and only if it is graphically strictly differentiable, i.e., the graph of the multifunction locally coincides, up to a change of coordinates, with the graph of a single-valued mapping, which is strictly differentiable at the transformed reference point. This result allows point-based characterizations of strict proto-differentiability in terms of various generalized derivatives. Further we will prove that under strict proto-differentiability the properties of strong metric regularity, metric regularity and strong metric subregularity are equivalent. Finally, under strict proto-differentiability of the subgradient mapping, we provide a novel second-order relation between function values and subgradients for prox-regular functions which constitutes a nonsmooth extension of the trapezoidal rule of numerical integration. |
| title | On strict proto-differentiability of set-valued mappings |
| topic | Optimization and Control 49J53, 90C31 |
| url | https://arxiv.org/abs/2411.01346 |