{
  "abstract": "We disprove a three-step error estimate for the Broyden--Fletcher--Goldfarb--Shanno {(BFGS)} method with exact line searches. We construct a globally strongly convex function $F\\in C^\\infty(\\mathbb R^3)$ and a single nonterminating trajectory $x_k\\to0$ for which $\\limsup_{k\\to\\infty}\\|x_{k+3}\\|/(\\|x_{k+2}\\|\\|x_k\\|)=+\\infty$. The Hessian of $F$ lies between $\\tfrac12 I$ and $\\tfrac32 I$ everywhere, and $F(x)-\\tfrac12\\|x\\|^2$ is compactly supported with all derivatives vanishing at $0$. The proof first constructs point-gradient pairs satisfying the exact BFGS step identities. A five-step cycle produces a large three-step quotient and restores an invariant plane for the actual updated matrix, allowing the cycle to be repeated with arbitrarily small gradient errors. Disjoint smooth corrections to a quadratic then realize all the pairs by one objective. Strong convexity identifies the prescribed points with the unique exact line-search minimizers. Thus smoothness and strong convexity do not imply the proposed product estimate, even for an objective that agrees with a quadratic to every order at its minimizer.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.02719",
  "author_total": 2,
  "authors": [
    {
      "name": "Jiawei Xu"
    },
    {
      "name": "Jiang Hu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/anjeyqdg28z2r.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.02719",
  "fulltext_url": null,
  "id": "anjeyqdg28z2r",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.02719",
  "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",
      "math.OC"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.02719",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/anjeyqdg28z2r.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:10Z",
      "native_id": "2611.02719",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.02719"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "A counterexample to a three-step convergence bound for BFGS with exact line search",
  "title_text": "A counterexample to a three-step convergence bound for BFGS with exact line search",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/anjeyqdg28z2r",
  "venue": "arXiv",
  "versions": []
}