{
  "abstract": "The classical Projected Hessian Lemma, originating from Finsler's theorem, characterizes definiteness over a constraint null space through quadratic penalties. However, it does not quantify the corresponding constrained eigenvalues or the associated eigenvalue penalty path. This work develops a quantitative penalty theory for constrained symmetric eigenvalue problems and establishes exact characterizations of the extremal eigenvalues of the reduced Hessian via full-space penalized eigenvalue problems. Three proofs are provided based on orthogonal decomposition, Schur complement analysis, and semidefinite programming duality. We characterize finite exact recovery along the extremal-eigenvalue penalty paths and, in the absence of finite recovery, establish asymptotic convergence with a first-order error expansion and an explicit leading coefficient. The associated Hellmann--Feynman sensitivity relation leads to a strategy for predicting penalty parameters. Based on these results, we develop a matrix-free Penalty--Split--Merge method for successive constrained extremal eigenpairs using penalty continuation, Split--Merge iterations, deflation, and projected certification. Numerical experiments illustrate the predicted penalty regimes, evaluate projected certification, and assess computational performance on moderate-scale benchmarks and large-scale matrix-free test instances.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.18538",
  "author_total": 2,
  "authors": [
    {
      "name": "Meiling Wang"
    },
    {
      "name": "Yong Xia"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0kdznzs9tns2q.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.18538",
  "fulltext_url": null,
  "id": "0kdznzs9tns2q",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.18538",
  "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",
      "1703|Matrix Theory and Algorithms"
    ]
  ],
  "openalex": "W7213512275",
  "pdf_url": "https://arxiv.org/pdf/2609.18538",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0kdznzs9tns2q.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:04Z",
      "native_id": "2609.18538",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.18538"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "The Projected Hessian Quantification Theorem: Exact Duality For Constrained Eigenvalues",
  "title_text": "The Projected Hessian Quantification Theorem: Exact Duality For Constrained Eigenvalues",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:04Z",
  "url": "https://research.draconitic.com/paper/0kdznzs9tns2q",
  "venue": "arXiv",
  "versions": []
}