Proximal Limited-Memory Quasi-Newton Methods for Nonsmooth Nonconvex Optimization
Abstract
We propose a regularized proximal quasi-Newton method (RPQN) for minimizing the sum of a continuously differentiable, possibly nonconvex function and a proper, lower semicontinuous, prox-bounded, possibly nonconvex regularizer. The method uses adaptive quadratic regularization and limited-memory quasi-Newton models. We establish global convergence under weak assumptions, without requiring global Lipschitz continuity of the gradient, and prove convergence of the entire sequence, with rates, under the Kurdyka-Lojasiewicz property. To solve the proximal subproblems efficiently, we exploit compact limited-memory representations and derive a compact representation of the Kleinmichel rank-one update, which preserves positive definiteness under the same condition as BFGS. Numerical experiments on logistic and Student's regression demonstrate competitive performance and the effectiveness of the proposed approach.
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BibTeX
@misc{dahl2026proximal,
title = {{Proximal Limited-Memory Quasi-Newton Methods for Nonsmooth Nonconvex Optimization}},
author = {Simeon vom Dahl and Alberto De Marchi and Christian Kanzow},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2605.11627},
eprint = {2605.11627},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.11627},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Proximal Limited-Memory Quasi-Newton Methods for Nonsmooth Nonconvex Optimization
AU - Simeon vom Dahl
AU - Alberto De Marchi
AU - Christian Kanzow
PY - 2026
DA - 2026/10/25
PB - arXiv
DO - 10.48550/arxiv.2605.11627
UR - https://arxiv.org/abs/2605.11627
AB - We propose a regularized proximal quasi-Newton method (RPQN) for minimizing the sum of a continuously differentiable, possibly nonconvex function and a proper, lower semicontinuous, prox-bounded, possibly nonconvex regularizer. The method uses adaptive quadratic regularization and limited-memory quasi-Newton models. We establish global convergence under weak assumptions, without requiring global Lipschitz continuity of the gradient, and prove convergence of the entire sequence, with rates, under the Kurdyka-Lojasiewicz property. To solve the proximal subproblems efficiently, we exploit compact limited-memory representations and derive a compact representation of the Kleinmichel rank-one update, which preserves positive definiteness under the same condition as BFGS. Numerical experiments on logistic and Student's regression demonstrate competitive performance and the effectiveness of the proposed approach.
ER -