Dimensionality Reduction of Affine Variational Inequalities Using Random Projections

TL;DR

This paper introduces a randomized dimensionality reduction method for affine variational inequalities using Johnson Lindenstrauss projections, enabling efficient approximate solutions with theoretical guarantees.

Contribution

It proposes a novel randomized algorithm that reduces the dimensionality of AVIs, providing a fast approximation method with proven accuracy and potential as an initialization subroutine.

Findings

The method achieves accurate approximations at significantly lower dimensions.

Numerical experiments confirm the theoretical guarantees and efficiency gains.

The approach reduces computational time for solving AVIs substantially.

Abstract

We present a method for dimensionality reduction of an affine variational inequality (AVI) defined over a compact feasible region. Centered around the Johnson Lindenstrauss lemma, our method is a randomized algorithm that produces with high probability an approximate solution for the given AVI by solving a lower-dimensional AVI. The algorithm allows the lower dimension to be chosen based on the quality of approximation desired. The algorithm can also be used as a subroutine in an exact algorithm for generating an initial point close to the solution. The lower-dimensional AVI is obtained by appropriately projecting the original AVI on a randomly chosen subspace. The lower-dimensional AVI is solved using standard solvers and from this solution an approximate solution to the original AVI is recovered through an inexpensive process. Our numerical experiments corroborate the theoretical results and validate that the algorithm provides a good approximation at low dimensions and substantial savings in time for an exact solution.