On the role of semismoothness in nonsmooth numerical analysis: Theory
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866908878458847232 |
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| author | Gfrerer, H. Outrata, J. V. |
| author_facet | Gfrerer, H. Outrata, J. V. |
| contents | For the numerical solution of nonsmooth problems, sometimes it is not necessary that an exact subgradient/generalized Jacobian is at our disposal, but it suffices that a semismooth derivative, i.e., a mapping satisfying a certain semismoothness property, is available. In this paper we consider not only semismooth derivatives of single-valued mappings, but also its interplay with the semismoothness$^*$ property for multifunctions. In particular, we are interested in semismooth derivatives of solution maps to parametric semismooth$^*$ inclusions. Our results are expressed in terms of suitable generalized derivatives of the set-valued part, i.e., by limiting coderivatives or by SC (subspace containing) derivatives. Further we show that semismooth derivatives coincide a.e. with generalized Jacobians and state some consequences concerning strict proto-differentiability for semismooth$^*$ multifunctions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_14637 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the role of semismoothness in nonsmooth numerical analysis: Theory Gfrerer, H. Outrata, J. V. Optimization and Control 65K10, 65K15, 90C26 For the numerical solution of nonsmooth problems, sometimes it is not necessary that an exact subgradient/generalized Jacobian is at our disposal, but it suffices that a semismooth derivative, i.e., a mapping satisfying a certain semismoothness property, is available. In this paper we consider not only semismooth derivatives of single-valued mappings, but also its interplay with the semismoothness$^*$ property for multifunctions. In particular, we are interested in semismooth derivatives of solution maps to parametric semismooth$^*$ inclusions. Our results are expressed in terms of suitable generalized derivatives of the set-valued part, i.e., by limiting coderivatives or by SC (subspace containing) derivatives. Further we show that semismooth derivatives coincide a.e. with generalized Jacobians and state some consequences concerning strict proto-differentiability for semismooth$^*$ multifunctions. |
| title | On the role of semismoothness in nonsmooth numerical analysis: Theory |
| topic | Optimization and Control 65K10, 65K15, 90C26 |
| url | https://arxiv.org/abs/2405.14637 |