{
  "abstract": "Proximal-gradient methods are widely used for composite optimization, but inexact proximal updates introduce a residual in the proximal optimality condition, complicating the analysis of how stationarity changes across a step. We study this question for composite objectives consisting of a smooth convex term and a possibly nonsmooth weakly convex term, focusing on the relationship between the directional-stationarity measures at the input and output points of an inexact proximal-gradient step. We first establish separate input and output bounds, but their combination can yield a conservative estimate because their worst-case residual directions may differ. By exploiting the magnitude and orientation of this shared residual, we establish tight joint bounds and derive an exact one-step stationarity-transfer factor that strictly improves the estimate from separate bounds in the short-step regime. We also identify the precise residual-tolerance threshold for finite uniform stationarity transfer. A further distinction emerges over multiple steps: endpoint worst cases can repeat over arbitrarily many iterations, whereas interior worst cases cannot persist across two consecutive steps, yielding strict two-step submultiplicativity. These results also provide principled tolerance-selection rules, descent estimates, and stationarity-complexity guarantees for inexact proximal-gradient methods.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.38310",
  "author_total": 1,
  "authors": [
    {
      "name": "Shenghan Mei"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/enrxs4x985va2.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.38310",
  "fulltext_url": null,
  "id": "enrxs4x985va2",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.38310",
  "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",
      "math.OC"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.38310",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/enrxs4x985va2.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2610.38310",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.38310"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Tight Stationarity Transfer and Repeatability for Inexact Proximal-Gradient Methods",
  "title_text": "Tight Stationarity Transfer and Repeatability for Inexact Proximal-Gradient Methods",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/enrxs4x985va2",
  "venue": "arXiv",
  "versions": []
}