{
  "abstract": "We resolve the open problem of deriving an exact, closed-form algebraic formula for the calmness modulus of $\\mathcal{C}^2$ convex constrained feasible set mappings $\\mathcal{F}(b) := \\{ x \\in \\mathbb{R}^n \\colon f_t(x) \\le b_t, \\, \\forall t \\in T \\}$ under right-hand side perturbations at a given pair of the graph $(\\bar{b}, \\bar{x})$. First, under the condition $0_n \\in \\mathrm{int}\\left(\\mathrm{conv}\\left\\{ \\nabla f_t(\\bar{x}) ,\\; t \\in T(\\bar{x}) \\right\\}\\right)$, we prove that the calmness modulus coincides with the inverse distance from the origin to the boundary of the nominal subdifferential. Second, for general $\\mathcal{C}^2$ convex systems where the previously mentioned interior condition does not hold, we introduce a second-order admissibility filtering over the active indexes. We show that this filtered family, $\\mathcal{D}^0_2(\\bar{x})$, yields an exact characterization of the calmness modulus relying solely on nominal data.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.06224",
  "author_total": 1,
  "authors": [
    {
      "name": "Jesús Camacho"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1brye15ejjt8r.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.06224",
  "fulltext_url": null,
  "id": "1brye15ejjt8r",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.06224",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2611.06224",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1brye15ejjt8r.ris",
  "sources": [
    {
      "first_seen": "2026-11-05T15:21:00Z",
      "native_id": "2611.06224",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.06224"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Calmness Modulus of Parameterized Convex Constraint Systems: Exact Closed-Form Formulas via Second-Order Admissibility",
  "title_text": "Calmness Modulus of Parameterized Convex Constraint Systems: Exact Closed-Form Formulas via Second-Order Admissibility",
  "type": "preprint",
  "updated_at": "2026-11-09T00:20:30Z",
  "url": "https://research.draconitic.com/paper/1brye15ejjt8r",
  "venue": "arXiv",
  "versions": []
}