{
  "abstract": "Critical Lagrange multipliers are known to be responsible for the potentially slow convergence of Newton-type methods, and noncriticality of a given multiplier has been tied to an error bound which estimates the distance of a primal-dual pair associated with a perturbed problem to the primal solution and the multiplier set of the unperturbed one, and is decisive for a local convergence analysis. We investigate this relation in an abstract setting where merely a set-valued mapping assigning primal-dual pairs to a parameter is available, and characterize the error bound of interest in terms of noncriticality of the underlying multiplier and two calmness-type properties of the associated multiplier mapping, one of which is a novel weak inner calmness in the fuzzy sense. No structural assumptions on the underlying problem are needed for that, and the two calmness-type conditions are not only sufficient but also necessary. Afterwards, we specify these findings for composite optimization problems. Whenever the subdifferential of the outer function is a polyhedral mapping, both conditions hold automatically, which explains the equivalence of noncriticality and the error bound observed in the polyhedral setting. Whenever the outer function is $C^2$-decomposable, they are secured by calmness of a restricted multiplier mapping together with a closedness condition, which recovers and extends the known characterization for $C^2$-cone reducible constraint systems.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.11491",
  "author_total": 2,
  "authors": [
    {
      "name": "Matúš Benko"
    },
    {
      "name": "Patrick Mehlitz"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0bfaa1e5qtgkw.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.11491",
  "fulltext_url": null,
  "id": "0bfaa1e5qtgkw",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.11491",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.OC"
    ],
    [
      "openalex",
      "1703|Optimization and Variational Analysis"
    ]
  ],
  "openalex": "W7221152331",
  "pdf_url": "https://arxiv.org/pdf/2610.11491",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0bfaa1e5qtgkw.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2610.11491",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.11491"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "On error bounds and noncritical Lagrange multipliers",
  "title_text": "On error bounds and noncritical Lagrange multipliers",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/0bfaa1e5qtgkw",
  "venue": "arXiv",
  "versions": []
}