Projected Subgradient Methods for a Class of Nonsmooth and Nonconvex Optimization Problems
Abstract
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-𝒞2 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.
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BibTeX
@misc{kanzow2026projected,
title = {{Projected Subgradient Methods for a Class of Nonsmooth and Nonconvex Optimization Problems}},
author = {Christian Kanzow and Jannis Krüger and Leo Lehmann},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.02517},
eprint = {2609.02517},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.02517},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Projected Subgradient Methods for a Class of Nonsmooth and Nonconvex Optimization Problems
AU - Christian Kanzow
AU - Jannis Krüger
AU - Leo Lehmann
PY - 2026
DA - 2026/09/02
PB - arXiv
DO - 10.48550/arxiv.2609.02517
UR - https://arxiv.org/abs/2609.02517
AB - 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-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.
ER -