The Geometry of Linear Program Compression: An Exact Characterization and Learning Algorithm

Abstract

We study how much a linear program (LP) can be compressed when solved repeatedly, given prior knowledge about its objective function. Existing data-driven projection methods learn low-dimensional surrogate LPs with approximate objective-value guarantees, but cannot provably identify the optimal projection for a prescribed compression budget. We instead ask a sharper question: how far can an LP be compressed into a lower-dimensional equivalent while \emph{exactly} preserving optimality, enabling faster repeated solves with no loss in solution quality? We provide an exact geometric characterization of such compressed LPs, together with a tractable sample-based learning algorithm that comes with fast-rate guarantees: the compressed LP recovers the optimal solution of an unseen instance with probability at least $1-\tilde{O}(d^{\star }/n)$, where $d^{\star }$ is the dimension of the decision-relevant subspace, and $n$ is the number of available historical LP samples. This $1/n$ dependence is sharper than the $\tilde{O}(1/\sqrt{n})$ uniform-convergence rates of approximate projection methods. Our framework further exposes a tunable tradeoff between the dimension of the compressed LP and the probability of recovering the optimal solution, allowing the user to trade compression for accuracy.

27 pages, 11 figures