Cone Rayleigh Levels: Finite Perturbations and Certified Control

Abstract

We study the reuse of positive trial profiles under finite perturbations of nonsymmetric matrix pencils B-λG. The lower and upper cone Rayleigh levels need not coincide and are defined without requiring positive eigenvectors. In the positive orthant with positive diagonal $G$, we derive computable perturbation bounds and determine the exact worst-case trial gap over prescribed independent entrywise perturbation classes. This yields the largest uniform radius meeting a given trial-gap tolerance for fixed profiles, certifying trial-value accuracy without recomputation. Experiments on a nonnegative operator and five signed matrices compare sufficient and optimal radii, directional thresholds, and cold- and warm-start recomputation. Several signed cases exhibit severely limited uniform profile reuse despite the optimality of the radius. Exact rational checks verify the reported bounds and worst-case constructions for stored numerical inputs. A finite-budget linear program and differentiable constraints illustrate applications to verified control and learning.

arXiv comment28 pages, 2 figures. Substantially revised and retitled. Added exact worst-case trial-gap formulas, optimal uniform perturbation radii, profile-preservation experiments, and common post-intervention profile refinement. Reproducibility package: https://doi.org/10.5281/zenodo.23150973