Tilt stability of Ky-Fan $κ$-norm composite optimization
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866910922061119488 |
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| author | Liu, Yulan Pan, Shaohua Song, Wen |
| author_facet | Liu, Yulan Pan, Shaohua Song, Wen |
| contents | This paper concerns the tilt stability for the minimization of the sum of a twice continuously differentiable matrix-valued function and the Ky-Fan $κ$-norm. To achieve this goal, we first provide a sufficient and necessary condition for a local minimizer of the composite $f=φ+g$ to be tilt-stable with the second subderivative of $g$, where $g$ is a closed proper convex function, and $φ$ is a twice continuously differentiable function that is locally convex at the local minimizer. Then, we apply the sufficient and necessary condition to the concerned Ky-Fan $κ$-norm composite problem, and employ the expression of second subderivative of the Ky-Fan $κ$-norm to derive a verifiable criterion to identify the tilt stability of a local minimum for this class of nonconvex and nonsmooth problems. As a byproduct, a practical criterion is obtained for identifying the tilt stablity of solutions to the nuclear-norm and spectral norm regularized minimization problems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_10945 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Tilt stability of Ky-Fan $κ$-norm composite optimization Liu, Yulan Pan, Shaohua Song, Wen Optimization and Control 49J52, 49J53, 49K40, 90C31, 90C56 This paper concerns the tilt stability for the minimization of the sum of a twice continuously differentiable matrix-valued function and the Ky-Fan $κ$-norm. To achieve this goal, we first provide a sufficient and necessary condition for a local minimizer of the composite $f=φ+g$ to be tilt-stable with the second subderivative of $g$, where $g$ is a closed proper convex function, and $φ$ is a twice continuously differentiable function that is locally convex at the local minimizer. Then, we apply the sufficient and necessary condition to the concerned Ky-Fan $κ$-norm composite problem, and employ the expression of second subderivative of the Ky-Fan $κ$-norm to derive a verifiable criterion to identify the tilt stability of a local minimum for this class of nonconvex and nonsmooth problems. As a byproduct, a practical criterion is obtained for identifying the tilt stablity of solutions to the nuclear-norm and spectral norm regularized minimization problems. |
| title | Tilt stability of Ky-Fan $κ$-norm composite optimization |
| topic | Optimization and Control 49J52, 49J53, 49K40, 90C31, 90C56 |
| url | https://arxiv.org/abs/2406.10945 |