Algorithms for Deciding Membership in Polytopes of General Dimension

Abstract: We study the fundamental problem of polytope membership aiming at large convex polytopes, i.e. in high dimension and with many facets, given as an intersection of halfspaces. Standard data-structures as well as brute force methods cannot scale, due to the curse of dimen- sionality. We design an efficient algorithm, by reduction to the approx- imate Nearest Neighbor (ANN) problem based on the construction of a Voronoi diagram with the polytope being one bounded cell. We thus trade exactness for efficiency so as to obtain complexity bounds polyno- mial in the dimension, by exploiting recent progress in the complexity of ANN search. We employ this algorithm to present a novel boundary data structure based on a Newton-like iterative intersection procedure. We implement our algorithms and compare with brute-force approaches to show that they scale very well as the dimension and number of facets grow larger.