{
  "abstract": "Given two quadratic functions \\( f(x) = x^T Ax + 2a^T x + a_0 \\) and \\( g(x) = x^T Bx + 2b^T x + b_0 ,\\) each associated with either the strict inequality ($<0$); non-strict inequality ($\\leq 0$); or the equality ($=0$), it is a fundamental question to ask whether or not the joint system has a solution. For homogeneous quadratic systems ($a=b=0,~a_0=b_0=0$), starting from Finsler's lemma in 1936 until Yuan's alternative lemma in 1990, all combinations of the unsolvability for $\\{x\\in \\mathbb{R} ^n\\mid x^T Ax \\mathbin{\\star } 0\\}\\cap \\{x\\in \\mathbb{R} ^n\\mid x^T Bx \\mathbin{\\#} 0\\}\\subset \\{0\\}$, where $\\star $ and $\\#$ can be any of $\\{<,\\leq ,=\\}$, have been shown to possess either a positive definite or a positive semi-definite matrix pencil of $A$ and $B.$ Extensions to nonhomogeneous quadratic systems $\\{x\\in \\mathbb{R} ^n\\mid f(x) \\mathbin{\\star } 0\\}\\cap \\{x\\in \\mathbb{R} ^n\\mid g(x) \\mathbin{\\#} 0\\}=\\emptyset $ have been done for several cases already. Two challenging cases remain open: the nonhomogeneous Calabi Theorem which determines when $\\{f(x)=0\\}\\cap \\{g(x)=0\\}=\\emptyset $; and the nonhomogeneous (strict) Finsler lemma to determine whether $\\{f(x)\\leq 0\\}\\cap \\{g(x)=0\\}=\\emptyset .$ The paper provides the answers to both, in theorems and algorithms.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.30571",
  "author_total": 3,
  "authors": [
    {
      "name": "Min-Chi Wang"
    },
    {
      "name": "Ruey-Lin Sheu"
    },
    {
      "name": "Huu-Quang Nguyen"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/19685jyv8b4fe.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.30571",
  "fulltext_url": null,
  "id": "19685jyv8b4fe",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.30571",
  "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": "W7204862048",
  "pdf_url": "https://arxiv.org/pdf/2608.30571",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-31",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/19685jyv8b4fe.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:48Z",
      "native_id": "2608.30571",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.30571"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Last two pieces of the puzzle for unsolvability of a system of two quadratic (in)equalities",
  "title_text": "Last two pieces of the puzzle for unsolvability of a system of two quadratic (in)equalities",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:48Z",
  "url": "https://research.draconitic.com/paper/19685jyv8b4fe",
  "venue": "arXiv",
  "versions": []
}