{
  "abstract": "We study trust-region minimization of a smooth, possibly nonconvex functional plus a convex functional composed with a bounded linear operator. A relative primal--dual gap condition controls both the error in an approximate proximal-gradient step and its linear-model decrease. Together with a computable absolute stationarity test, it yields a finite Cauchy search, convergence of the proximal stationarity measure to zero, and an $O(\\varepsilon^{-2})$ bound on outer trials. The outer analysis allows the linear operator to take values in a Banach space and does not require dual attainment. When the operator takes values in a Hilbert space and the regularizer is finite and Lipschitz, the dual proximal-gradient method produces finite gaps tending to zero, provided the required proximal maps and functional values can be evaluated. We prove $O(j^{-1})$ gap bounds for both recovered and averaged primal candidates and give a sharper bound on the primal error for exactly recovered points. A semilinear elliptic control problem with unsmoothed total-variation regularization and an $L^2$ control cost illustrates the method in the full $H^1$ metric. Across five meshes, outer and state Newton counts remain constant, while interior-point iteration counts vary mildly.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.27348",
  "author_total": 1,
  "authors": [
    {
      "name": "Harbir Antil"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2dw76aykkwqd3.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.27348",
  "fulltext_url": null,
  "id": "2dw76aykkwqd3",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.27348",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.NA"
    ],
    [
      "arxiv",
      "math.AP"
    ],
    [
      "arxiv",
      "math.NA"
    ],
    [
      "arxiv",
      "math.OC"
    ],
    [
      "openalex",
      "1703|Optimization and Variational Analysis"
    ]
  ],
  "openalex": "W7214189478",
  "pdf_url": "https://arxiv.org/pdf/2609.27348",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2dw76aykkwqd3.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:00Z",
      "native_id": "2609.27348",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.27348"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Relative Primal--Dual Gap Certificates for Operator-Composite Trust-Region Methods",
  "title_text": "Relative Primal--Dual Gap Certificates for Operator-Composite Trust-Region Methods",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:00Z",
  "url": "https://research.draconitic.com/paper/2dw76aykkwqd3",
  "venue": "arXiv",
  "versions": []
}