A randomized algorithm for enumerating zonotope vertices

Abstract: We propose a randomized algorithm for enumerating the vertices of a zonotope, which is a low-dimensional linear projection of a hypercube. The algorithm produces a pair of the zonotope's vertices by sampling a random linear combination of the zonotope generators, where the combination's weights are the signs of the product between the zonotope's generator matrix and random vectors with normally distributed entries. We study the probability of recovering particular vertices and relate it to the vertices' normal cones. This study shows that if we terminate the randomized algorithm before all vertices are recovered, then the convex hull of the resulting vertex set approximates the zonotope. In high dimensions, we expect the enumeration algorithm to be most appropriate as an approximation algorithm---particularly for cases when existing methods are not practical.