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| Autori principali: | , |
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| Natura: | Preprint |
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2026
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| Soggetti: | |
| Accesso online: | https://arxiv.org/abs/2605.12666 |
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| _version_ | 1866917488467378176 |
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| author | Bodard, Alexander Patrinos, Panagiotis |
| author_facet | Bodard, Alexander Patrinos, Panagiotis |
| contents | Newton-type methods are typically analyzed under Lipschitz continuity of the Hessian, an assumption that can fail for objectives with higher-order or polynomial growth. We introduce a class of nonlinearly preconditioned Newton methods that apply Newton's root-finding scheme to a transformed optimality mapping, thereby extending recent nonlinear preconditioning ideas from first-order methods to the second-order setting. The resulting methods are naturally analyzed under Lipschitz continuity of a preconditioned Hessian, a condition that significantly relaxes the classical Hessian Lipschitz continuity assumption. Under this generalized smoothness model, we establish local superlinear and quadratic convergence guarantees, and develop a globalization strategy for the nonregularized method despite the fact that the preconditioned Newton direction need not be a descent direction. We further propose a regularized variant for isotropic preconditioners, and show that it attains an $O(\varepsilon^{-3/2})$ iteration complexity. An adaptive version removes the need to know the smoothness constant and allows inexact subproblem solutions while preserving the same complexity order. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_12666 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Newton methods beyond Hessian Lipschitz continuity: A nonlinear preconditioning approach Bodard, Alexander Patrinos, Panagiotis Optimization and Control Newton-type methods are typically analyzed under Lipschitz continuity of the Hessian, an assumption that can fail for objectives with higher-order or polynomial growth. We introduce a class of nonlinearly preconditioned Newton methods that apply Newton's root-finding scheme to a transformed optimality mapping, thereby extending recent nonlinear preconditioning ideas from first-order methods to the second-order setting. The resulting methods are naturally analyzed under Lipschitz continuity of a preconditioned Hessian, a condition that significantly relaxes the classical Hessian Lipschitz continuity assumption. Under this generalized smoothness model, we establish local superlinear and quadratic convergence guarantees, and develop a globalization strategy for the nonregularized method despite the fact that the preconditioned Newton direction need not be a descent direction. We further propose a regularized variant for isotropic preconditioners, and show that it attains an $O(\varepsilon^{-3/2})$ iteration complexity. An adaptive version removes the need to know the smoothness constant and allows inexact subproblem solutions while preserving the same complexity order. |
| title | Newton methods beyond Hessian Lipschitz continuity: A nonlinear preconditioning approach |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2605.12666 |