Smoothness of Subgradient Mappings and Its Applications in Parametric Optimization
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866914996010614784 |
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| author | Hang, Nguyen T. V. Sarabi, Ebrahim |
| author_facet | Hang, Nguyen T. V. Sarabi, Ebrahim |
| contents | We demonstrate that the concept of strict proto-differentiability of subgradient mappings can play a similar role as smoothness of the gradient mapping of a function in the study of subgradient mappings of prox-regular functions. We then show that metric regularity and strong metric regularity are equivalent for a class of generalized equations when this condition is satisfied. For a class of composite functions, called C2-decomposable, we argue that strict proto-differentiability can be characterized via a simple relative interior condition. Leveraging this observation, we present a characterization of the continuous differentiability of the proximal mapping for this class of function via a certain relative interior condition. Applications to the study of strong metric regularity of the KKT system of a class of composite optimization problems are also provided. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2311_06026 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Smoothness of Subgradient Mappings and Its Applications in Parametric Optimization Hang, Nguyen T. V. Sarabi, Ebrahim Optimization and Control 90C31, 65K99, 49J52, 49J53 We demonstrate that the concept of strict proto-differentiability of subgradient mappings can play a similar role as smoothness of the gradient mapping of a function in the study of subgradient mappings of prox-regular functions. We then show that metric regularity and strong metric regularity are equivalent for a class of generalized equations when this condition is satisfied. For a class of composite functions, called C2-decomposable, we argue that strict proto-differentiability can be characterized via a simple relative interior condition. Leveraging this observation, we present a characterization of the continuous differentiability of the proximal mapping for this class of function via a certain relative interior condition. Applications to the study of strong metric regularity of the KKT system of a class of composite optimization problems are also provided. |
| title | Smoothness of Subgradient Mappings and Its Applications in Parametric Optimization |
| topic | Optimization and Control 90C31, 65K99, 49J52, 49J53 |
| url | https://arxiv.org/abs/2311.06026 |