{
  "abstract": "We propose Broximal Gradient Descent (BroxGD), a projection-free sister method to projected gradient descent for constrained optimization. Its forward--backward construction replaces the proximal backward operation by the broximal operation of Gruntkowska et al. (2025). Instead of projecting, each step minimizes a linear function over the intersection of the constraint set $\\mathcal{X}$ and a ball $\\mathbb{B}(x_k,t_k)$ centered at the current iterate $x_k$, of suitable radius $t_k>0$: \\[ x_{k+1}\\in\\arg\\min_{z\\in\\mathcal{X}\\cap\\mathbb{B}(x_k,t_k)}\\langle\\nabla f(x_k),z\\rangle. \\] We develop a comprehensive convergence theory spanning a wide range of optimization regimes and radius rules. We expect BroxGD to find many applications and inspire numerous extensions, much like projected gradient descent. Our contribution is theoretical; potential applications and toy experiments illustrate the method and suggest directions for future work.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2605.08850",
  "author_total": 3,
  "authors": [
    {
      "name": "Peter Richtárik"
    },
    {
      "name": "Kaja Gruntkowska"
    },
    {
      "name": "Hanmin Li"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/a8ewb598hp7kk.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2605.08850",
  "fulltext_url": null,
  "id": "a8ewb598hp7kk",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2605.08850",
  "language": null,
  "licence": "CC0-1.0",
  "licence_label": "CC0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/publicdomain/zero/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.LG"
    ],
    [
      "arxiv",
      "math.OC"
    ],
    [
      "arxiv",
      "stat.ML"
    ],
    [
      "openalex",
      "1703|Optimization and Variational Analysis"
    ]
  ],
  "openalex": "W7160887272",
  "pdf_url": "https://arxiv.org/pdf/2605.08850",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/a8ewb598hp7kk.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2605.08850",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2605.08850"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ML",
      "subfield": "statistics"
    }
  ],
  "title": "Broximal Gradient Descent: A Projection-Free Sister of Projected Gradient Descent",
  "title_text": "Broximal Gradient Descent: A Projection-Free Sister of Projected Gradient Descent",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:26Z",
  "url": "https://research.draconitic.com/paper/a8ewb598hp7kk",
  "venue": "arXiv",
  "versions": []
}