Asymptotic confidence sets for random linear programs

TL;DR

This paper develops computationally efficient methods to derive asymptotic confidence sets for solutions of random linear programs, with applications to statistical analysis of optimal transport problems.

Contribution

It introduces a new, tractable expression for the distributional limits of linear program solutions, improving upon previous intractable formulations.

Findings

New polynomial-time computable expression for distributional limits

Distributional limits for entire solution sets when optima are not unique

A data-driven method for constructing asymptotic confidence sets

Abstract

Motivated by the statistical analysis of the discrete optimal transport problem, we prove distributional limits for the solutions of linear programs with random constraints. Such limits were first obtained by Klatt, Munk, & Zemel (2022), but their expressions for the limits involve a computationally intractable decomposition of $\mathbb{R}^m$ into a possibly exponential number of convex cones. We give a new expression for the limit in terms of auxiliary linear programs, which can be solved in polynomial time. We also leverage tools from random convex geometry to give distributional limits for the entire set of random optimal solutions, when the optimum is not unique. Finally, we describe a simple, data-driven method to construct asymptotically valid confidence sets in polynomial time.