The Nearest Polytope Problem: Algorithms and Application to Controlling Hybrid Systems

We present three algorithms for solving the nearest polytope problem: given a list of polytopes and a distance metric in Euclidean space, find the nearest polytope to a query point. We consider the AH-polytope representation, which generalizes the H-polytope representation and is particularly useful in the context of control. Through preprocessing the polytopes into efficient data structures, we avoid exhaustive search at query time. We discuss the properties of the algorithms and compare their performances using polytopic datasets motivated by control applications, including sampling-based motion planning and model predictive control.