{
  "abstract": "We study the Bregman proximal point method for the obstacle problem, a fundamental variational inequality arising in contact mechanics, optimal design, and mathematical finance. Each Bregman proximal step regularizes the energy through a Bregman divergence generated by a Legendre function, leading to a semilinear elliptic subproblem. We first establish a well-posedness and strict feasibility theory for these subproblems. We then analyze the convergence of the resulting iteration in the $H^1$-norm. Our convergence analysis is based on a sequential strict local minimality inequality. We show abstractly how the order of minimality determines the convergence rate. Assuming that the initial guess lies above the exact solution and a one-sided Bregman growth condition holds, we establish sequential strict local minimality of order $s \\geq 2$ and derive sublinear convergence rates in the $H^1$-norm of the form $O(k^{-ζ})$, where $ζ\\in(1/2,1]$ depends on the choice of Legendre function. For the Shannon and Tsallis entropies, we show that these rates are sharp in a uniform worst-case sense. Under additional assumptions on the free boundary and the obstacle, we establish sequential strict local minimality of order $s = 1$ with the Shannon and Spence entropies. This yields linear convergence of the form $O(\\varrho^k)$ for some $\\varrho\\in(0,1)$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.23007",
  "author_total": 3,
  "authors": [
    {
      "name": "Brendan Keith"
    },
    {
      "name": "Haojun Qin"
    },
    {
      "name": "Noe Reyes Rivas"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3gefqdqdfb6em.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.23007",
  "fulltext_url": null,
  "id": "3gefqdqdfb6em",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.23007",
  "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.AP"
    ],
    [
      "openalex",
      "1703|Optimization and Variational Analysis"
    ]
  ],
  "openalex": "W7214030731",
  "pdf_url": "https://arxiv.org/pdf/2609.23007",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3gefqdqdfb6em.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2609.23007",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.23007"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Analysis of Bregman Proximal Point for the Obstacle Problem",
  "title_text": "Analysis of Bregman Proximal Point for the Obstacle Problem",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/3gefqdqdfb6em",
  "venue": "arXiv",
  "versions": []
}