Practical Volume Estimation by a New Annealing Schedule for Cooling Convex Bodies

Abstract

We study the problem of estimating the volume of convex polytopes, focusing on H- and V-polytopes, as well as zonotopes. Although a lot of effort is devoted to practical algorithms for H-polytopes there is no such method for the latter two representations. We propose a new, practical algorithm for all representations, which is faster than existing methods. It relies on Hit-and-Run sampling, and combines a new simulated annealing method with the Multiphase Monte Carlo (MMC) approach. Our method introduces the following key features to make it adaptive: (a) It defines a sequence of convex bodies in MMC by introducing a new annealing schedule, whose length is shorter than in previous methods with high probability, and the need of computing an enclosing and an inscribed ball is removed; (b) It exploits statistical properties in rejection-sampling and proposes a better empirical convergence criterion for specifying each step; (c) For zonotopes, it may use a sequence of convex bodies for MMC different than balls, where the chosen body adapts to the input. We offer an open-source, optimized C++ implementation, and analyze its performance to show that it outperforms state-of-the-art software for H-polytopes by Cousins-Vempala (2016) and Emiris-Fisikopoulos (2018), while it undertakes volume computations that were intractable until now, as it is the first polynomial-time, practical method for V-polytopes and zonotopes that scales to high dimensions (currently 100). We further focus on zonotopes, and characterize them by their order (number of generators over dimension), because this largely determines sampling complexity. We analyze a related application, where we evaluate methods of zonotope approximation in engineering.Comment: 20 pages, 12 figures, 3 table