{
  "abstract": "We introduce the Unified Separation Condition (USC) - a simple and general constraint qualification for finite-dimensional nonlinear programming. The USC requires that the origin be uniformly separated from the subdifferential of the constraint violation function in a neighborhood of the feasible set. The main result of this paper is two-fold. First, we show that the classical Mangasarian--Fromovitz constraint qualification (MFCQ) for inequalities and the linear independence constraint qualification (LICQ) for equalities imply the USC. Conversely, USC is strictly more general: it remains applicable in nonsmooth settings where classical conditions are not defined, and it may hold even when MFCQ fails. The proof that MFCQ implies USC is based on Gordan's theorem; the implication from LICQ to USC follows from the linear independence of the gradients of the active constraints. We also introduce a local version of USC and discuss its computational verification via convex quadratic programming. The USC provides a unified framework for constraint qualifications, bridging classical smooth theory and nonsmooth optimization.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.16981",
  "author_total": 1,
  "authors": [
    {
      "name": "Majid E. Abbasov"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/161c3f7tw07xt.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.16981",
  "fulltext_url": null,
  "id": "161c3f7tw07xt",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.16981",
  "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/2609.16981",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/161c3f7tw07xt.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:45Z",
      "native_id": "2609.16981",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.16981"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "The Unified Separation Condition: A General Constraint Qualification for Smooth and Nonsmooth Optimization",
  "title_text": "The Unified Separation Condition: A General Constraint Qualification for Smooth and Nonsmooth Optimization",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:45Z",
  "url": "https://research.draconitic.com/paper/161c3f7tw07xt",
  "venue": "arXiv",
  "versions": []
}