We consider an affine variational inequality posed over a polyhedral convex set in n‐dimensional Euclidean space. It is often the case that this underlying set has dimension less than n, or has a nontrivial lineality space, or both. We show that when the variational inequality satisfies a well known regularity condition, we can reduce the problem to the solution of an affine variational inequality in a space of smaller dimension, followed by some simple linear‐algebraic calculations. The smaller problem inherits the regularity condition from the original one, and therefore it has a unique solution. The dimension of the space in which the smaller problem is posed equals the rank of the original set: that is, its dimension less the dimension of the lineality space.
