{
  "abstract": "Whether the Broyden-Fletcher-Goldfarb-Shanno (BFGS) method and its variants can fail to converge on smooth nonconvex functions under realistic line-search parameters has remained a central open problem in quasi-Newton theory since the landmark counterexample of Dai (2002), which was confined to small Armijo parameters ($c_1 \\le 1/84 \\approx 0.0119$ on Powell's geometry, or $c_1 \\le 69/7480 \\approx 0.0092$ on his six-point cycle) and an objective function unbounded below. In this paper, we resolve the long-standing open question, posed by Dai (2002) following a discussion with J. C. Gilbert, of whether such counterexamples exist in theory for every Armijo parameter $c_1 \\in (0, 1)$. Specifically, for every prescribed pair of line-search parameters $0 < c_1 < c_2 < 1$, we construct an objective function $f \\in C^\\infty(\\mathbb{R}^2)$, bounded below and with Lipschitz continuous gradient, on which every method in a broad conjugacy class $\\mathcal{C}$ equipped with the first-local-minimizer line search generates an infinite sequence of iterates with $\\|\\nabla f(x_k)\\| = 1$ for all $k \\ge 0$. The class $\\mathcal{C}$ encompasses the classical full-memory BFGS method, limited-memory BFGS (L-BFGS) with arbitrary memory $m \\ge 1$, the Broyden positive family, and the Hestenes-Stiefel conjugate gradient method. The steps are the standard first local minimizers along the search rays and simultaneously satisfy the strong Wolfe, weak Wolfe, Armijo, and Goldstein conditions with constants $(c_1, c_2)$. The construction operates in the minimal possible dimension $n = 2$, exploiting a non-decaying conjugate descent orbit in the plane coupled with an explicit tubular interpolation whose two-bump axial curvature profile places the Armijo ratio anywhere in $(0, 1)$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.09686",
  "author_total": 1,
  "authors": [
    {
      "name": "Rui Diao"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/17v40jhjgnxgt.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.09686",
  "fulltext_url": null,
  "id": "17v40jhjgnxgt",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.09686",
  "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",
      "2612|Advanced Optimization Algorithms Research"
    ]
  ],
  "openalex": "W7212107007",
  "pdf_url": "https://arxiv.org/pdf/2609.09686",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/17v40jhjgnxgt.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:15Z",
      "native_id": "2609.09686",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.09686"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Counterexamples for BFGS-type methods under arbitrary strong Wolfe constants",
  "title_text": "Counterexamples for BFGS-type methods under arbitrary strong Wolfe constants",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:15Z",
  "url": "https://research.draconitic.com/paper/17v40jhjgnxgt",
  "venue": "arXiv",
  "versions": []
}