arXiv subjects

Simeon vom Dahl

Publications and source records attributed to Simeon vom Dahl.

3 recordsLinked to original sources

arXiv2026

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

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.

math.OC↗

arXiv2024

An Inexact Regularized Proximal Newton Method without Line Search

In this paper, we introduce an inexact regularized proximal Newton method (IRPNM) that does not require any line search. The method is designed to minimize the sum of a twice continuously differentiable function $f$ and a convex (possibly non-smooth and extended-valued) function $φ$. Instead of controlling a step size by a line search procedure, we update the regularization parameter in a suitable way, based on the success of the previous iteration. The global convergence of the sequence of iterations and its superlinear convergence rate under a local Hölderian error bound assumption are shown. Notably, these convergence results are obtained without requiring a global Lipschitz property for $ \nabla f $, which, to the best of the authors' knowledge, is a novel contribution for proximal Newton methods. To highlight the efficiency of our approach, we provide numerical comparisons with an IRPNM using a line search globalization and a modern FISTA-type method.

math.OC↗

arXiv2020

Solving polyhedral d.c. optimization problems via concave minimization

The problem of minimizing the difference of two convex functions is called polyhedral d.c. optimization problem if at least one of the two component functions is polyhedral. We characterize the existence of global optimal solutions of polyhedral d.c. optimization problems. This result is used to show that, whenever the existence of an optimal solution can be certified, polyhedral d.c. optimization problems can be solved by certain concave minimization algorithms. No further assumptions are necessary in case of the first component being polyhedral and just some mild assumptions to the first component are required for the case where the second component is polyhedral. In case of both component functions being polyhedral, we obtain a primal and dual existence test and a primal and dual solution procedure. Numerical examples are discussed.

math.OC↗