Proximal Limited-Memory Quasi-Newton Methods for Nonsmooth Nonconvex Optimization

Simeon vom Dahl; Alberto De Marchi; Christian Kanzow

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arXiv · 2605.11627

Proximal Limited-Memory Quasi-Newton Methods for Nonsmooth Nonconvex Optimization

Simeon vom Dahl·Alberto De Marchi·Christian Kanzow

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Abstract

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.

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BibTeX RIS

Simeon vom Dahl, Alberto De Marchi, Christian Kanzow. 2026-05-12. Proximal Limited-Memory Quasi-Newton Methods for Nonsmooth Nonconvex Optimization. https://arxiv.org/abs/2605.11627

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