Random projection of Linear and Semidefinite problem with linear inequalities

TL;DR

This paper extends the Johnson-Lindenstrauss lemma to inequality constraints in linear and semidefinite programs, enabling dimension reduction and constraint aggregation with theoretical guarantees on solution quality.

Contribution

Introduces a new random matrix with non-negative entries for aggregating inequality constraints in LPs, extending previous results to a broader class of problems.

Findings

The method preserves the optimal value approximately.

Reduces the number of constraints and problem dimension.

Extends to certain semidefinite programming problems.

Abstract

The Johnson-Lindenstrauss Lemma states that there exist linear maps that project a set of points of a vector space into a space of much lower dimension such that the Euclidean distance between these points is approximately preserved. This lemma has been previously used to prove that we can randomly aggregate, using a random matrix whose entries are drawn from a zero-mean sub-Gaussian distribution, the equality constraints of an Linear Program (LP) while preserving approximately the value of the problem. In this paper we extend these results to the inequality case by introducing a random matrix with non-negative entries that allows to randomly aggregate inequality constraints of an LP while preserving approximately the value of the problem. By duality, the approach we propose allows to reduce both the number of constraints and the dimension of the problem while obtaining some theoretical guarantees on the optimal value. We will also show an extension of our results to certain semidefinite programming instances.