{
  "abstract": "Singularity degree is an important measure of semidefinite programming (SDP) degeneracy, but it is generally unavailable a priori from the problem data. We augment the Shor relaxation of nonempty binary sets $\\{x\\in\\{0,1\\}^n:Ax=b\\}$ with the first-level Reformulation-Linearization Technique (RLT) equations generated by the defining linear equalities. For the resulting equality-generated SDP-RLT relaxation, we determine the exact worst-case singularity degree. If $\\operatorname{rank}(A)=m$ and $0<m<n$, then the associated relaxation has singularity degree at most $\\min\\{m,n-m\\}$, and this rank-nullity bound is attained for every possible rank in this range. Consequently, the worst-case singularity degree over this class is $\\lfloor n/2\\rfloor$ for $n\\geq 2$. This is strikingly smaller than the sharp general bound $n$ for feasible SDP systems with matrix variables of order $n+1$ (Sturm, 2000, Example 2). Thus, for individual relaxations, rank and nullity provide an a priori bound on the otherwise inaccessible singularity degree and on the Hölder exponent in error bounds estimating distance to feasibility from constraint residuals.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.29945",
  "author_total": 1,
  "authors": [
    {
      "name": "Hao Hu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/51e5mkryyhkj4.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.29945",
  "fulltext_url": null,
  "id": "51e5mkryyhkj4",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.29945",
  "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",
      "2612|Advanced Optimization Algorithms Research"
    ]
  ],
  "openalex": "W7204947624",
  "pdf_url": "https://arxiv.org/pdf/2608.29945",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/51e5mkryyhkj4.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:48Z",
      "native_id": "2608.29945",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.29945"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Sharp Singularity-Degree Bounds for Equality-Generated SDP-RLT Relaxations of Binary Programs",
  "title_text": "Sharp Singularity-Degree Bounds for Equality-Generated SDP-RLT Relaxations of Binary Programs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:48Z",
  "url": "https://research.draconitic.com/paper/51e5mkryyhkj4",
  "venue": "arXiv",
  "versions": []
}