Proximal Limited-Memory Quasi-Newton Methods for Nonsmooth Nonconvex Optimization
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| Format: | Preprint |
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2026
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| _version_ | 1866913115882389504 |
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| author | Dahl, Simeon vom De Marchi, Alberto Kanzow, Christian |
| author_facet | Dahl, Simeon vom De Marchi, Alberto Kanzow, Christian |
| contents | We introduce a proximal limited--memory quasi--Newton scheme for minimizing the sum of a continuously differentiable function and a proper, lower semicontinuous and prox-bounded, possibly nonsmooth, function. Both functions might be nonconvex. The method builds upon the computation of scaled proximal operators and is globalized by adaptively updating a regularization parameter based on a criterion of sufficient decrease. We prove global convergence under mild assumptions and then establish convergence of the entire sequence (with rates) under the Kurdyka--Lojasiewicz property. To efficiently solve the subproblems, we exploit the compact representation of limited-memory quasi-Newton updates. We derive also a compact representation of the limited--memory Kleinmichel formula, a rank-one quasi-Newton scheme that preserves positive definiteness under the same condition as the BFGS update. Numerical results show a significant speed up compared to other methods. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_11627 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Proximal Limited-Memory Quasi-Newton Methods for Nonsmooth Nonconvex Optimization Dahl, Simeon vom De Marchi, Alberto Kanzow, Christian Optimization and Control 65K10, 90C06, 90C26, 90C53 We introduce a proximal limited--memory quasi--Newton scheme for minimizing the sum of a continuously differentiable function and a proper, lower semicontinuous and prox-bounded, possibly nonsmooth, function. Both functions might be nonconvex. The method builds upon the computation of scaled proximal operators and is globalized by adaptively updating a regularization parameter based on a criterion of sufficient decrease. We prove global convergence under mild assumptions and then establish convergence of the entire sequence (with rates) under the Kurdyka--Lojasiewicz property. To efficiently solve the subproblems, we exploit the compact representation of limited-memory quasi-Newton updates. We derive also a compact representation of the limited--memory Kleinmichel formula, a rank-one quasi-Newton scheme that preserves positive definiteness under the same condition as the BFGS update. Numerical results show a significant speed up compared to other methods. |
| title | Proximal Limited-Memory Quasi-Newton Methods for Nonsmooth Nonconvex Optimization |
| topic | Optimization and Control 65K10, 90C06, 90C26, 90C53 |
| url | https://arxiv.org/abs/2605.11627 |