{
  "abstract": "We develop and analyze a Bregman-projected mirror iteration for low-order regularizations of stationary mean-field game (MFG) systems in their natural Banach space setting. For separable Hamiltonians of the form $H(x,p,m)=H_0(x,p)-g(m)$, with quadratic or super-quadratic Hamiltonian growth and linear or super-linear density couplings, we formulate a low-order $\\barγ$-Laplacian regularization of the stationary MFG system as a variational inequality on $L^{\\barβ}(\\mathbb T^d)\\times W^{1,\\barγ}(\\mathbb T^d)$. To approximate solutions of this regularized variational inequality, we introduce a Bregman geometry matched to the mixed Lebesgue-Sobolev exponents of the problem and analyze the exact constrained one-step mirror iteration and a frozen two-step variant. Under quantitative monotonicity of the coupling, both exact iterations converge strongly to the unique solution of the regularized variational inequality for each fixed $\\varepsilon>0$, under natural summability conditions on the step sizes. Along weakly convergent vanishing-regularization sequences, we also prove strong convergence in the full product space. Numerical experiments in one and two dimensions show first-order accuracy on manufactured solutions and weak dependence of outer iteration counts on the tested grids. At matched effective step sizes, the frozen two-step and one-step implementations agree up to inner-solve accuracy, while the latter requires fewer mirror solves.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2606.19611",
  "author_total": 4,
  "authors": [
    {
      "name": "Hussain Al Abdulaziz"
    },
    {
      "name": "Yuri Ashrafyan"
    },
    {
      "name": "Yeva Gevorgyan"
    },
    {
      "name": "Diogo Gomes"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2htte50x4sq2n.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2606.19611",
  "fulltext_url": null,
  "id": "2htte50x4sq2n",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2606.19611",
  "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",
      "cs.NA"
    ],
    [
      "arxiv",
      "math.AP"
    ],
    [
      "arxiv",
      "math.NA"
    ],
    [
      "openalex",
      "1703|Optimization and Variational Analysis"
    ]
  ],
  "openalex": "W7165392605",
  "pdf_url": "https://arxiv.org/pdf/2606.19611",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2htte50x4sq2n.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2606.19611",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2606.19611"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "Bregman-projected mirror methods for regularized stationary mean-field games",
  "title_text": "Bregman-projected mirror methods for regularized stationary mean-field games",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/2htte50x4sq2n",
  "venue": "arXiv",
  "versions": []
}