{
  "abstract": "We analyze a one-evaluation-per-iteration method for $\\mathrm{VI}(C,B)$ with monotone $L$-Lipschitz $B$: $x_{k+1}=P_C(x_k-λB(u_k))$, $u_k=x_k+θ_k(x_k-x_{k-1})+β_k(x_k-u_{k-1})$, $θ_k+β_k=1$. For constant step and summable filter ($\\sum_kβ_k<\\infty$) we prove weak convergence via a Lyapunov function with exact rational dissipation budgets, including Malitsky's reflected gradient method. It is robust to summable operator errors; if $C$ is bounded, the constant-step range reaches $λ<(\\sqrt2-1)/L$. For unbounded $C$ this reduces to an a priori boundedness statement, certified to $λL=0.387$ by dissipation trading; for affine $B$ on polyhedral $C$, $\\dist(x_n,S)\\to0$ with $\\sum_n\\dist^2(x_n,S)<\\infty$, strong convergence for bounded $C$, and $R$-linear rates after face identification. The main result is a safeguarded adaptive step-size rule needing no $L$ and no extra evaluations, proved weakly convergent unconditionally: a data-driven Lyapunov weight removes $L$ from the dissipation budgets. For affine $B$ with $C=\\mathcal H$, sharpness of $λL=1/\\sqrt3$ via a rotation lower bound; the same constant is sharp for unconstrained nonlinear $B$, with convergence for square-summable operator values. Unconditional convergence below $1/(\\sqrt3\\,L)$ reduces to a marginal-pole absolute-stability statement; the projected case remains open. Under strong monotonicity we prove $R$-linear convergence with explicit contraction, and certify $Ω(L)$ speedups over constant steps. Numerics confirm the gains.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.18355",
  "author_total": 1,
  "authors": [
    {
      "name": "Yekini Shehu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/egzy5znnz3xm9.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.18355",
  "fulltext_url": null,
  "id": "egzy5znnz3xm9",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.18355",
  "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": "W7213502663",
  "pdf_url": "https://arxiv.org/pdf/2609.18355",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/egzy5znnz3xm9.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:04Z",
      "native_id": "2609.18355",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.18355"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "A Parameter-Free Adaptive Reflected Gradient Method for Monotone Variational Inequalities",
  "title_text": "A Parameter-Free Adaptive Reflected Gradient Method for Monotone Variational Inequalities",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:04Z",
  "url": "https://research.draconitic.com/paper/egzy5znnz3xm9",
  "venue": "arXiv",
  "versions": []
}