The Projected Hessian Quantification Theorem: Exact Duality For Constrained Eigenvalues

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.