{
  "abstract": "To minimize the sum $f + g + h$ of three convex functions when $f$ is differentiable with a Lipschitz-continuous gradient, we present a proximal algorithm, variable-metric three operator splitting (VMTOS), a generalization of the three operator splitting (TOS) scheme of Davis and Yin. We establish convergence guarantees by controlling changes in the metric. The smallest saddle-point residual among the first $k$ primal-dual iterates is $\\tilde{O}(k^{-1/2})$. When $f$ and the convex conjugate $h^*$ are strongly convex and the accepted metrics and their inverses are uniformly bounded, the rate improves to $\\tilde{O}(β^k)$ for some $β\\in (0,1)$. We develop a practical Barzilai--Borwein update for diagonal metrics that satisfies our bound on metric changes. Under the same strong convexity assumptions, this update guarantees the required uniform metric bounds, yielding R-linear convergence. Numerical experiments on benchmark optimization problems show that VMTOS is competitive with other proximal algorithms.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.38935",
  "author_total": 2,
  "authors": [
    {
      "name": "Hansol Suh"
    },
    {
      "name": "Tobin Isaac"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6sg8jwh76ng5t.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.38935",
  "fulltext_url": null,
  "id": "6sg8jwh76ng5t",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.38935",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2610.38935",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6sg8jwh76ng5t.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2610.38935",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.38935"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Variable Metric Three Operator Splitting",
  "title_text": "Variable Metric Three Operator Splitting",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/6sg8jwh76ng5t",
  "venue": "arXiv",
  "versions": []
}