arXiv subjects

Jannis Krüger

Publications and source records attributed to Jannis Krüger.

2 recordsLinked to original sources

arXiv2026

Projected Subgradient Methods for a Class of Nonsmooth and Nonconvex Optimization Problems

We investigate the optimization problem of minimizing a nonsmooth function that satisfies a nonsmooth version of the descent lemma over a nonempty and closed but not necessarily convex set. The objective function belongs to the class of upper-$\mathcal{C}^2$ functions, whereas the constraints may promote a sparse or low-rank structure. We propose a projected subgradient method with two different globalization strategies: (a) a nonmonotone linesearch and, under additional assumptions, (b) an auto-conditioned method, where the stepsize is given by a formula depending on data from past iterations. We show that both methods converge to solutions that satisfy a stronger stationarity concept than one would expect from the subdifferential sum-rule, which is particularly important since the optimization problems of interest are inherently nonconvex. Finally, we present promising numerical results when applying the algorithm to an MPEC-style problem as well as the matrix optimization problems MAXCUT and Robust PCA.

math.OC↗

arXiv2026

Convergence of the Safeguarded Augmented Lagrangian Method under the Polyak-Lojasiewicz constraint qualification for Constrained Composite Optimization

In this work we provide theoretical and practical results of the Safeguarded Augmented Lagrangian Method (SALM) for constrained composite optimization problems whose objective is the sum of a smooth and a nonsmooth function. We obtain global convergence results to an M-stationary point for SALM under the Polyak-Lojasiewicz constraint qualification (PLCQ). For this result the boundedness of the Lagrange multipliers is crucial, and this is shown under the assumption that the nonsmooth part of the objective function is locally Lipschitz continuous. A counterexample shows that one cannot expect to get bounded multipliers without such an assumption. The performance of the algorithm is evaluated numerically on a set of sparse portfolio optimization problems with two different regularization terms, one being Lipschitz and the other one being non-Lipschitz. The results are significantly better for the Lipschitz sparsity term, whereas the underlying method generates seemingly unbounded multipliers in many instances when using the non-Lipschitz sparsity function.

math.OC↗