Column-Randomized Linear Programs: Performance Guarantees and Applications

TL;DR

This paper introduces a randomized sampling method for solving large-scale linear programs with many columns, providing probabilistic guarantees on solution quality and demonstrating effectiveness in practical applications.

Contribution

It proposes a novel column sampling approach with theoretical performance bounds, offering an alternative to traditional column generation methods.

Findings

The method achieves an optimality gap decreasing at a rate of 1/√K.

Theoretical bounds relate the gap to a distributional linear program.

Numerical experiments validate effectiveness in cutting-stock and choice modeling.

Abstract

We propose a randomized method for solving linear programs with a large number of columns but a relatively small number of constraints. Since enumerating all the columns is usually unrealistic, such linear programs are commonly solved by column generation, which is often still computationally challenging due to the intractability of the subproblem in many applications. Instead of iteratively introducing one column at a time as in column generation, our proposed method involves sampling a collection of columns according to a user-specified randomization scheme and solving the linear program consisting of the sampled columns. While similar methods for solving large-scale linear programs by sampling columns (or, equivalently, sampling constraints in the dual) have been proposed in the literature, in this paper we derive an upper bound on the optimality gap that holds with high probability. This bound converges at a rate $1 / \sqrt{K}$, where $K$ is the number of sampled columns, to the optimality gap of a linear program related to the sampling distribution. We analyze the gap of this latter linear program, which we dub the distributional counterpart, and derive conditions under which this gap will be small. Finally, we numerically demonstrate the effectiveness of the proposed method in the cutting-stock problem and in nonparametric choice model estimation.