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The Projected Hessian Quantification Theorem: Exact Duality For Constrained Eigenvalues — arXiv:2609.18538

Source: arXiv (en)
Original: https://arxiv.org/abs/2609.18538
License (as archived): CC0-1.0
Archived: 2026-10-15T15:40:21.871Z
SHA-256 of the archived text: 397f079911bf5e15973e9a507c3f0d0d4b26536edc10885f655263bb46ef767a
Stable URL of this page: https://metraipolis.com/documento/dc_e22da221adbb

How to cite

The Projected Hessian Quantification Theorem: Exact Duality For Constrained Eigenvalues — arXiv:2609.18538 — arXiv (en), https://arxiv.org/abs/2609.18538 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_e22da221adbb (Metraipolis archive dc_e22da221adbb, sha256 397f079911bf5e15973e9a507c3f0d0d4b26536edc10885f655263bb46ef767a)

BibTeX

@misc{dce22da221adbb,
  title = {The Projected Hessian Quantification Theorem: Exact Duality For Constrained Eigenvalues — arXiv:2609.18538},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2609.18538},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_e22da221adbb; sha256 397f079911bf5e15973e9a507c3f0d0d4b26536edc10885f655263bb46ef767a},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2609.18538. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

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.

Tema degli esploratori: matematica e informatica teorica (retro-catalogo arXiv: annunci del 2026-09-17 in math.OC).
Categorie arXiv: Optimization and Control (math.OC).
Pagina del lavoro: https://arxiv.org/abs/2609.18538 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-10-15T15:08:22.598Z dalla pagina https://arxiv.org/catchup/math.OC/2026-09-17?abs=True (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).