{
  "abstract": "We give an $R$-linear convergence proof for BB1, BB2, and fixed positive spectral-weight variants on finite-dimensional strictly convex quadratics. The delayed recurrence is written as a first-order system on the closed cone of compatible consecutive-gradient pairs. Its transition is continuous and positively homogeneous of degree one, including at finite termination. A compactness theorem shows that pointwise convergence of every cone orbit implies a uniform finite-step contraction and hence a global $R$-linear estimate. By a realization lemma, every compatible state initiates a BB1 trajectory, so Raydan's theorem yields pointwise stability on the entire cone. Finally, the transformation $g\\mapsto W^{1/2}g$ conjugates every fixed positive spectral-weight rule to BB1. Consequently, BB1, BB2, and all such weighted rules have the same homogeneous growth radius.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.12100",
  "author_total": 2,
  "authors": [
    {
      "name": "Shutai Yang"
    },
    {
      "name": "Ya-xiang Yuan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/17bcp6c9rqpj7.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.12100",
  "fulltext_url": null,
  "id": "17bcp6c9rqpj7",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.12100",
  "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",
      "cs.NA"
    ],
    [
      "arxiv",
      "math.NA"
    ],
    [
      "arxiv",
      "math.OC"
    ],
    [
      "openalex",
      "2612|Advanced Optimization Algorithms Research"
    ]
  ],
  "openalex": "W7212651442",
  "pdf_url": "https://arxiv.org/pdf/2609.12100",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-10",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/17bcp6c9rqpj7.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:11Z",
      "native_id": "2609.12100",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.12100"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "$R$-Linear Convergence of Barzilai-Borwein Methods on Strictly Convex Quadratics via Closed-Cone Homogeneous Dynamics",
  "title_text": "R-Linear Convergence of Barzilai-Borwein Methods on Strictly Convex Quadratics via Closed-Cone Homogeneous Dynamics",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:11Z",
  "url": "https://research.draconitic.com/paper/17bcp6c9rqpj7",
  "venue": "arXiv",
  "versions": []
}