  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.
