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| Formato: | Preprint |
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2024
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| Acceso en línea: | https://arxiv.org/abs/2412.09898 |
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| _version_ | 1866913719446929408 |
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| author | He, Jiahuan Kan, Chao Song, Wen |
| author_facet | He, Jiahuan Kan, Chao Song, Wen |
| contents | In this paper, our focus lies on the study of the second-order variational analysis of orthogonally invariant matrix functions. It is well-known that an orthogonally invariant matrix function is an extended-real-value function defined on ${\mathbb M}_{m,n}\,(n \leqslant m)$ of the form $f \circ σ$ for an absolutely symmetric function $f \colon \R^n \rightarrow [-\infty,+\infty]$ and the singular values $σ\colon {\mathbb M}_{m,n} \rightarrow \R^{n}$. We establish several second-order properties of orthogonally invariant matrix functions, such as parabolic epi-differentiability, parabolic regularity, and twice epi-differentiability when their associated absolutely symmetric functions enjoy some properties. Specifically, we show that the nuclear norm of a real $m \times n$ matrix is twice epi-differentiable and we derive an explicit expression of its second-order epi-derivative. Moreover, for a convex orthogonally invariant matrix function, we calculate its second subderivative and present sufficient conditions for twice epi-differentiability. This enables us to establish second-order optimality conditions for a class of matrix optimization problems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_09898 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Twice Epi-Differentiability of Orthogonally Invariant Matrix Functions and Application He, Jiahuan Kan, Chao Song, Wen Optimization and Control In this paper, our focus lies on the study of the second-order variational analysis of orthogonally invariant matrix functions. It is well-known that an orthogonally invariant matrix function is an extended-real-value function defined on ${\mathbb M}_{m,n}\,(n \leqslant m)$ of the form $f \circ σ$ for an absolutely symmetric function $f \colon \R^n \rightarrow [-\infty,+\infty]$ and the singular values $σ\colon {\mathbb M}_{m,n} \rightarrow \R^{n}$. We establish several second-order properties of orthogonally invariant matrix functions, such as parabolic epi-differentiability, parabolic regularity, and twice epi-differentiability when their associated absolutely symmetric functions enjoy some properties. Specifically, we show that the nuclear norm of a real $m \times n$ matrix is twice epi-differentiable and we derive an explicit expression of its second-order epi-derivative. Moreover, for a convex orthogonally invariant matrix function, we calculate its second subderivative and present sufficient conditions for twice epi-differentiability. This enables us to establish second-order optimality conditions for a class of matrix optimization problems. |
| title | Twice Epi-Differentiability of Orthogonally Invariant Matrix Functions and Application |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2412.09898 |