{
  "abstract": "We consider the Frank--Wolfe algorithm for solving variational inequalities over compact polytopes under a monotone $C^1$ operator $Φ$. The linear minimization oracle returns vertices, and we assume a coercivity condition on the Jacobian of $Φ$ at a solution. We prove last-iterate convergence to the unique solution at rate $O(k^{-\\min\\{α,1\\}})$ for step sizes $γ_k=α/(k+β)$ with $β\\geqα>0$. The bound holds uniformly over starting points and vertex tie-breaking. In particular, both $1/(k+1)$ and $2/(k+2)$ give $O(k^{-1})$ distance. A matching lower bound shows that this distance guarantee is worst-case order-optimal among prescribed admissible step-size schedules. We also obtain rates $O(k^{-\\min\\{α,1\\}})$ for the Frank--Wolfe gap and $O(k^{-\\min\\{α,2\\}})$ for the dual gap. These yield $O(\\varepsilon^{-1})$ and $O(\\varepsilon^{-1/2})$ calls to each oracle for sustained gap accuracy $\\varepsilon$ when $α\\geq1$ and $α\\geq2$, respectively, with instance-dependent constants. We also treat more general prescribed step-size schedules.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.17127",
  "author_total": 1,
  "authors": [
    {
      "name": "Tobias Harks"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7fathvdnq0hre.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.17127",
  "fulltext_url": null,
  "id": "7fathvdnq0hre",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.17127",
  "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": "W7221824657",
  "pdf_url": "https://arxiv.org/pdf/2610.17127",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7fathvdnq0hre.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:19Z",
      "native_id": "2610.17127",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.17127"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Last-Iterate Convergence of Frank--Wolfe for Nonlinear Variational Inequalities on Polytopes",
  "title_text": "Last-Iterate Convergence of Frank--Wolfe for Nonlinear Variational Inequalities on Polytopes",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:19Z",
  "url": "https://research.draconitic.com/paper/7fathvdnq0hre",
  "venue": "arXiv",
  "versions": []
}