{
  "abstract": "The Barzilai--Borwein (BB) method is $R$-linearly convergent on strongly convex quadratics, although neither the objective value nor the gradient norm is generally monotone. We construct an explicit current-point Lyapunov function for the quadratic dynamics after the warm-up step, assuming that both endpoints of the initial active spectrum survive this step. If $a<b$ are these endpoints and $P_a,P_b$ are the corresponding spectral projectors, then \\[ \\overline{f}(x) :=\\lVert P_a\\nabla f(x)\\rVert^{\\frac{2b}{a+b}} \\lVert P_b\\nabla f(x)\\rVert^{\\frac{2a}{a+b}} \\] satisfies, for BB1, BB2, and every fixed positive weighted delayed Rayleigh rule, \\[ \\overline{f}(x_{k+1}) =\\left(\\frac{b-a}{b+a}\\right)^2 e^{-2D_{a,b}(τ_k)}\\overline{f}(x_k), \\qquad D_{a,b}(τ_k)\\ge0, \\] where $τ_k=1/α_k$ is the reciprocal step. The function is obtained by denormalizing the endpoint coboundary certificate in the sharp-rate analysis of Yang and Yuan. We prove that the endpoint exponents are the unique minimax choice among endpoint monomials and illustrate the Lyapunov law on a nonmonotone BB1 trajectory. Using this law, we also give a proof of $R$-linear convergence of the method in finite dimensions.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.12084",
  "author_total": 2,
  "authors": [
    {
      "name": "Shutai Yang"
    },
    {
      "name": "Ya-xiang Yuan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6nv454dzv8a0c.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.12084",
  "fulltext_url": null,
  "id": "6nv454dzv8a0c",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.12084",
  "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": "W7212652804",
  "pdf_url": "https://arxiv.org/pdf/2609.12084",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-10",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6nv454dzv8a0c.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:11Z",
      "native_id": "2609.12084",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.12084"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Lyapunov Functions and $R$-Linear Convergence for Quadratic Barzilai-Borwein Dynamics",
  "title_text": "Lyapunov Functions and R-Linear Convergence for Quadratic Barzilai-Borwein Dynamics",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:11Z",
  "url": "https://research.draconitic.com/paper/6nv454dzv8a0c",
  "venue": "arXiv",
  "versions": []
}