{
  "abstract": "The dual active-set method of Goldfarb and Idnani solves the strictly convex quadratic program by adding and dropping one constraint at a time, requiring no phase one. Primal--dual active set and block principal pivoting skip the walk altogether: they guess an entire active set at once and repair it from returning signs. For bound constraints the guess is safe. The system solved on a candidate set is a principal submatrix of $G^{-1}$, positive definite regardless of the guess; this $P$-matrix property guarantees finite termination. For $C^\\top x \\ge b$ that hypothesis fails, which is our central focus. The working-set system is $C_{\\mathcal{A}}^\\top G^{-1} C_{\\mathcal{A}}$, positive definite only when $C_{\\mathcal{A}}$ has full column rank, a property of the guess, not the data. The $P$-matrix property is lost and the structural obstruction invalidates the guarantee. Strict convexity keeps the method usable: KKT conditions make candidate sets certified rather than trusted. Over 600 random instances across four constraint families this failure never occurs. However, duplicating columns of $C$ raises failure rates to 93\\%, dropping certified fraction from 100\\% to zero. On real long-only portfolio data, the active set reaches 442 of $494 + 1$ constraints.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.30933",
  "author_total": 2,
  "authors": [
    {
      "name": "Thomas Schmelzer"
    },
    {
      "name": "Martin Stoll"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dew2rs9ahbj4q.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.30933",
  "fulltext_url": null,
  "id": "dew2rs9ahbj4q",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.30933",
  "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"
    ],
    [
      "openalex",
      "2612|Advanced Optimization Algorithms Research"
    ]
  ],
  "openalex": "W7204967744",
  "pdf_url": "https://arxiv.org/pdf/2608.30933",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-31",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dew2rs9ahbj4q.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:48Z",
      "native_id": "2608.30933",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.30933"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Goldfarb-Idnani Revisited:Invariants, Certificates, and the Limits of Guessing",
  "title_text": "Goldfarb-Idnani Revisited:Invariants, Certificates, and the Limits of Guessing",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:48Z",
  "url": "https://research.draconitic.com/paper/dew2rs9ahbj4q",
  "venue": "arXiv",
  "versions": []
}