{
  "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-$\\mathcal{C}^2$ 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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.02517",
  "author_total": 3,
  "authors": [
    {
      "name": "Christian Kanzow"
    },
    {
      "name": "Jannis Krüger"
    },
    {
      "name": "Leo Lehmann"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2vjtgefb3gykq.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.02517",
  "fulltext_url": null,
  "id": "2vjtgefb3gykq",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.02517",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.OC"
    ],
    [
      "openalex",
      "1703|Optimization and Variational Analysis"
    ]
  ],
  "openalex": "W7207824053",
  "pdf_url": "https://arxiv.org/pdf/2609.02517",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2vjtgefb3gykq.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:20Z",
      "native_id": "2609.02517",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.02517"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Projected Subgradient Methods for a Class of Nonsmooth and Nonconvex Optimization Problems",
  "title_text": "Projected Subgradient Methods for a Class of Nonsmooth and Nonconvex Optimization Problems",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:20Z",
  "url": "https://research.draconitic.com/paper/2vjtgefb3gykq",
  "venue": "arXiv",
  "versions": []
}