A practical algorithm for volume estimation based on billiard trajectories and simulated annealing

TL;DR

This paper introduces a practical Monte Carlo algorithm using billiard trajectories and simulated annealing for efficient volume estimation of convex polytopes across different representations, enabling scalable computations in high dimensions.

Contribution

The paper presents a novel multiphase Monte Carlo method with empirical convergence tests and adaptive annealing, significantly improving volume approximation for high-dimensional polytopes.

Findings

Successfully scales to thousands of dimensions for H-polytopes

Achieves high accuracy for V- and Z-polytopes in moderate hardware

Provides the first software capable of handling such high-dimensional problems

Abstract

We tackle the problem of efficiently approximating the volume of convex polytopes, when these are given in three different representations: H-polytopes, which have been studied extensively, V-polytopes, and zonotopes (Z-polytopes). We design a novel practical Multiphase Monte Carlo algorithm that leverages random walks based on billiard trajectories, as well as a new empirical convergence tests and a simulated annealing schedule of adaptive convex bodies. After tuning several parameters of our proposed method, we present a detailed experimental evaluation of our tuned algorithm using a rich dataset containing Birkhoff polytopes and polytopes from structural biology. Our open-source implementation tackles problems that have been intractable so far, offering the first software to scale up in thousands of dimensions for H-polytopes and in the hundreds for V- and Z-polytopes on moderate hardware. Last, we illustrate our software in evaluating Z-polytope approximations.