{
  "abstract": "We study the scalar proximity operator of the nonconvex power penalty with exponent between zero and one. Lagrange--Bürmann inversion gives an exact series for the larger stationary root, and the series converges uniformly throughout the active domain of the proximity operator. For rational exponents, an explicit residue-class decomposition reorganizes the coefficients into finite sums of generalized hypergeometric functions, recovering the familiar special cases and producing an additional formula for \\(p=\\tfrac13\\). A Mellin--Barnes formula yields an equivalent residue-corrected representation and analytic truncation bounds. We distinguish the proximal activation threshold from the lower saddle-node threshold and prove that the active proximal branch is uniformly well-conditioned; the practical difficulty near activation is slow series convergence when the exponent approaches one, rather than singular root sensitivity. We also formulate an inexact proximal-gradient step using a computable optimality residual and a descent test. With summable descent errors and vanishing residuals, the objective values converge, the steps vanish, and every cluster point is critical. Tests over a broad parameter grid and in sparse-recovery proximal-gradient runs illustrate both evaluator accuracy and the implementability of the acceptance certificates.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2606.31951",
  "author_total": 2,
  "authors": [
    {
      "name": "Lixin Shen"
    },
    {
      "name": "Jiangyu Yu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5awf8dnw5j2q4.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2606.31951",
  "fulltext_url": null,
  "id": "5awf8dnw5j2q4",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2606.31951",
  "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",
      "1703|Optimization and Variational Analysis"
    ]
  ],
  "openalex": "W7166888203",
  "pdf_url": "https://arxiv.org/pdf/2606.31951",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5awf8dnw5j2q4.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:04Z",
      "native_id": "2606.31951",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2606.31951"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Exact Series and Error-Controlled Evaluation of the Nonconvex $\\ell_p$ Proximity Operator",
  "title_text": "Exact Series and Error-Controlled Evaluation of the Nonconvex ℓ_p Proximity Operator",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:04Z",
  "url": "https://research.draconitic.com/paper/5awf8dnw5j2q4",
  "venue": "arXiv",
  "versions": []
}