{
  "abstract": "We study convex quadratic minimization over the unit $L_1$ ball in which the maximum eigenvalue of the Hessian matrix is bounded by a positive constant $L$. We propose a novel first-order algorithm with objective value error bounded by $O(L/T^3)$ after $T$ gradient evaluations, assuming that the subproblems involved in the algorithm can be solved exactly. To the best of our knowledge, the best convergence rate of algorithms in the literature is $O(L/T^2)$. From the perspective of information-based complexity theory, our proposed algorithm is the first in the literature that achieves the $O((L/\\varepsilon)^{1/3})$ first-order oracle complexity, although its current version is not necessarily practical for implementation. We hope that our proposed algorithm could shed some light on future implementable and efficient $O(L/T^3)$-convergence-rate algorithms. The proposed algorithm incorporates a decomposition of components of vectors in the unit $L_1$-norm ball to \"good\" and \"bad\" parts, and uses symmetric rank-1 (SR1) updates on the bad parts. The proposed algorithm was developed after the author instructed the OpenAI ChatGPT 6 (Astra) model to study the problem using ideas of weak-type $L^1$ estimates and good-bad part decomposition in harmonic analysis and a recent result.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.10314",
  "author_total": 1,
  "authors": [
    {
      "name": "Yuyuan Ouyang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/b6ddaw6rcd6ac.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.10314",
  "fulltext_url": null,
  "id": "b6ddaw6rcd6ac",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.10314",
  "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",
      "2612|Advanced Optimization Algorithms Research"
    ]
  ],
  "openalex": "W7212134436",
  "pdf_url": "https://arxiv.org/pdf/2609.10314",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/b6ddaw6rcd6ac.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:15Z",
      "native_id": "2609.10314",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.10314"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "An $O(1/T^3)$ algorithm for minimizing convex quadratic functions over the $L_1$ ball",
  "title_text": "An O(1/T^3) algorithm for minimizing convex quadratic functions over the L_1 ball",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:15Z",
  "url": "https://research.draconitic.com/paper/b6ddaw6rcd6ac",
  "venue": "arXiv",
  "versions": []
}