# Column-Randomized Linear Programs: Performance Guarantees and Applications

39 Pages Posted: 10 Sep 2020

See all articles by Yi-Chun Chen

## Yi-Chun Chen

University of California, Los Angeles (UCLA) - Anderson School of Management

## Velibor Mišić

University of California, Los Angeles (UCLA) - Anderson School of Management

Date Written: July 20, 2020

### 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 and converges with rate $1/\sqrt{K}$, where $K$ is the number of sampled columns, to the value of a linear program related to the sampling distribution. To the best of our knowledge, this is the first paper addressing the convergence of the optimality gap for sampling columns/constraints in generic linear programs without additional assumptions on the problem structure and sampling distribution. We further apply the proposed method to various applications, such as linear programs with totally unimodular constraints, Markov decision processes, covering problems and packing problems, and derive problem-specific performance guarantees. We also generalize the method to the case that the sampled columns may not be statistically independent. Finally, we numerically demonstrate the effectiveness of the proposed method in the cutting-stock problem and in nonparametric choice model estimation.

Keywords: linear programming; constraint sampling

Suggested Citation

Chen, Yi-Chun and Misic, Velibor, Column-Randomized Linear Programs: Performance Guarantees and Applications (July 20, 2020). Available at SSRN: https://ssrn.com/abstract=3656704 or http://dx.doi.org/10.2139/ssrn.3656704