{
  "abstract": "We establish whole-sequence convergence of the primal iterates of the smoothing-based full-splitting proximal subgradient method (S-FSPS) of Boţ, Li, and Tao (SIAM J. Optim., 35 (2025), pp.~2623--2653) with a prescribed, nonsummable sequence of vanishing smoothing parameters. The challenge is that each iteration uses a different smoothed model. We address this by constructing fixed lifted potentials, deriving compatible sufficient-decrease and relative-error estimates, and applying a scaled Kurdyka--Łojasiewicz (KL) finite-length argument that controls the residuals caused by changes in the smoothing parameter. Under a KL assumption on the corresponding fixed lifted potential and a weighted summability condition on these residuals, the primal trajectory has finite length. For power schedules $γ_k=(k+k_0)^{-β}$, with $k_0\\geq1$ and $1/2<β\\leq1$, the summability condition holds when the lift exponent is sufficiently large. Consequently, the primal sequence converges to an exact limiting lifted stationary point without imposing full-row-rank assumptions on the linear operators. As a corollary, we establish whole-sequence convergence for a variable-smoothing full-splitting projected-gradient method in nonconvex nonsmooth composite optimization. Examples distinguish primal convergence from convergence of the auxiliary dual variables and illustrate the role of nonsummability in guaranteeing exact stationarity.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.28306",
  "author_total": 1,
  "authors": [
    {
      "name": "Min Tao"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/e9wrzkfxk5xze.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.28306",
  "fulltext_url": null,
  "id": "e9wrzkfxk5xze",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.28306",
  "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": "W7214173270",
  "pdf_url": "https://arxiv.org/pdf/2609.28306",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/e9wrzkfxk5xze.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:00Z",
      "native_id": "2609.28306",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.28306"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Whole-Sequence Convergence of Variable-Smoothing Full-Splitting Methods via a Lifted Kurdyka-Łojasiewicz Framework",
  "title_text": "Whole-Sequence Convergence of Variable-Smoothing Full-Splitting Methods via a Lifted Kurdyka-Łojasiewicz Framework",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:00Z",
  "url": "https://research.draconitic.com/paper/e9wrzkfxk5xze",
  "venue": "arXiv",
  "versions": []
}