{
  "abstract": "In this paper, we develop provable parameter-free fixed-point algorithms for contractive mappings, with an emphasis on automatically exploiting hidden contractivity without requiring prior knowledge of the contraction factor. Our first method is a parameter-free variant of the Halpern fixed-point iteration with no line-search, bisection, or prior estimate of the contraction factor, while retaining essentially the same per-iteration computational cost as classical fixed-point schemes. We establish explicit linear convergence rates for both the fixed-point residual and the distance to the unique fixed point. The second algorithm is also a parameter-free Halpern method, but it requires a line-search procedure to find a local estimate for the contraction factor, which may need a few additional operator evaluations. This method also enjoys explicit linear convergence guarantees. We further extend these ideas in two directions. First, using the relation between Halpern iterations and Nesterov-type accelerated fixed-point schemes, we derive parameter-free Nesterov-type (three-term) variants that inherit linear convergence in the contractive setting. Second, by combining the proposed fixed-point schemes with Tikhonov regularization, we obtain new methods for solving cocoercive equations and establish an iteration complexity of $\\mathcal{O}({ε^{-1}\\log(ε^{-1})})$ for computing an $ε$-solution, provided that a valid lower bound on the cocoercivity constant is available. Numerical experiments on several examples demonstrate that the proposed algorithms are competitive with, and often outperform, existing adaptive fixed-point methods. In particular, the methods successfully exploit contractive behavior when it is present on the selected nonexpansive test problems.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.09022",
  "author_total": 3,
  "authors": [
    {
      "name": "Quoc Tran-Dinh"
    },
    {
      "name": "Pham Ngoc Anh"
    },
    {
      "name": "Ha Manh Tien"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2ewjyzcpcfzw0.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.09022",
  "fulltext_url": null,
  "id": "2ewjyzcpcfzw0",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.09022",
  "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"
    ],
    [
      "arxiv",
      "stat.ML"
    ],
    [
      "openalex",
      "1703|Optimization and Variational Analysis"
    ]
  ],
  "openalex": "W7202172621",
  "pdf_url": "https://arxiv.org/pdf/2608.09022",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2ewjyzcpcfzw0.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:28:40Z",
      "native_id": "2608.09022",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.09022"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ML",
      "subfield": "statistics"
    }
  ],
  "title": "Provable Parameter-Free Fixed-Point Algorithms with Linear Convergence Rates",
  "title_text": "Provable Parameter-Free Fixed-Point Algorithms with Linear Convergence Rates",
  "type": "preprint",
  "updated_at": "2026-10-30T12:28:46Z",
  "url": "https://research.draconitic.com/paper/2ewjyzcpcfzw0",
  "venue": "arXiv",
  "versions": []
}