Random projections for linear programming

TL;DR

This paper demonstrates that random projections can be used to efficiently approximate solutions to large-scale linear programming problems by preserving key geometric properties, including distances to cones.

Contribution

It extends the Johnson-Lindenstrauss lemma to preserve distances to cones and introduces a probabilistic algorithm for approximate linear programming solutions.

Findings

The algorithm efficiently solves large random LP instances.

Random projections preserve distances to cones, not just Euclidean distances.

Application demonstrated in error correction coding problem.

Abstract

Random projections are random linear maps, sampled from appropriate distributions, that approx- imately preserve certain geometrical invariants so that the approximation improves as the dimension of the space grows. The well-known Johnson-Lindenstrauss lemma states that there are random ma- trices with surprisingly few rows that approximately preserve pairwise Euclidean distances among a set of points. This is commonly used to speed up algorithms based on Euclidean distances. We prove that these matrices also preserve other quantities, such as the distance to a cone. We exploit this result to devise a probabilistic algorithm to solve linear programs approximately. We show that this algorithm can approximately solve very large randomly generated LP instances. We also showcase its application to an error correction coding problem.