Dimensionality reduction of affine variational inequalities using random projections

We present a method for dimensionality reduction of an affine variational inequality (AVI) defined over a compact feasible region. Our method is a randomized algorithm centered around the Johnson Lindenstrauss lemma [1] that produces with high probability an approximate solution for the given AVI by solving a lower-dimensional AVI. The lower dimension can 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 obtained through an inexpensive process. Our numerical experiments corroborate the theoretical results and validate that the algorithm provides a good approximation at very low dimensions and substantial savings in time for an exact solution.