On the equivalence of a Hessian-free inequality and Lipschitz continuous Hessian
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| Main Authors: | , , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866918006733406208 |
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| author | Boţ, Radu I. Dao, Minh N. Liu, Tianxiang Lourenço, Bruno F. Marumo, Naoki |
| author_facet | Boţ, Radu I. Dao, Minh N. Liu, Tianxiang Lourenço, Bruno F. Marumo, Naoki |
| contents | It is known that if a twice differentiable function has a Lipschitz continuous Hessian, then its gradients satisfy a Jensen-type inequality. In particular, this inequality is Hessian-free in the sense that the Hessian does not actually appear in the inequality. In this paper, we show that the converse holds in a generalized setting: if a continuos function from a Hilbert space to a reflexive Banach space satisfies such an inequality, then it is Fréchet differentiable and its derivative is Lipschitz continuous. Our proof relies on the Baillon-Haddad theorem. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_17193 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the equivalence of a Hessian-free inequality and Lipschitz continuous Hessian Boţ, Radu I. Dao, Minh N. Liu, Tianxiang Lourenço, Bruno F. Marumo, Naoki Optimization and Control Functional Analysis It is known that if a twice differentiable function has a Lipschitz continuous Hessian, then its gradients satisfy a Jensen-type inequality. In particular, this inequality is Hessian-free in the sense that the Hessian does not actually appear in the inequality. In this paper, we show that the converse holds in a generalized setting: if a continuos function from a Hilbert space to a reflexive Banach space satisfies such an inequality, then it is Fréchet differentiable and its derivative is Lipschitz continuous. Our proof relies on the Baillon-Haddad theorem. |
| title | On the equivalence of a Hessian-free inequality and Lipschitz continuous Hessian |
| topic | Optimization and Control Functional Analysis |
| url | https://arxiv.org/abs/2504.17193 |