发表机构
School of System Science and Statistics, Beijing Wuzi University; LMIB of the Ministry of Education, School of Mathematical Sciences, Beihang University（北京物资大学系统科学与统计学院; 北京航空航天大学数学科学学院教育部重点实验室）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出投影Hessian量化定理，精确刻画约束特征值，并开发无矩阵罚-分裂-合并方法，用于高效求解约束极值特征对。
AI 中文摘要
经典的投影Hessian引理源自Finsler定理，通过二次罚函数刻画约束零空间上的正定性。然而，该引理并未量化相应的约束特征值或相关的特征值罚路径。本文针对约束对称特征值问题发展了定量罚理论，并通过全空间罚特征值问题建立了约化Hessian极值特征值的精确刻画。我们提供了三种证明，分别基于正交分解、Schur补分析和半定规划对偶性。我们刻画了沿极值特征值罚路径的有限精确恢复，并在缺乏有限恢复的情况下，建立了具有一阶误差展开和显式首项系数的新近收敛性。相关的Hellmann–Feynman灵敏度关系导出了预测罚参数的策略。基于这些结果，我们开发了一种无矩阵的罚-分裂-合并方法，通过罚连续、分裂-合并迭代、收缩和投影认证来连续求解约束极值特征对。数值实验展示了预测的罚区间，评估了投影认证，并在中等规模基准和大规模无矩阵测试实例上评估了计算性能。
英文摘要
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.
Comments20 pages, 1 figure
