Average-case thresholds for exact regularization of linear programs

Abstract: Small regularizers can preserve linear programming solutions exactly. This paper provides the first average-case analysis of exact regularization: with a standard Gaussian cost vector and fixed constraint set, bounds are established for the probability that exact regularization succeeds as a function of regularization strength. Failure is characterized via the Gaussian measure of inner cones, controlled by novel two-sided bounds on the measure of shifted cones. Results reveal dimension-dependent scaling laws and connect exact regularization of linear programs to their polyhedral geometry via the normal fan and the Gaussian (solid-angle) measure of its cones. Computable bounds are obtained in several canonical settings, including regularized optimal transport. Numerical experiments corroborate the predicted scalings and thresholds.