Practical Volume Estimation of Zonotopes 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. The number of phases in MMC tends to constant as order increases, while the bodies in (c) are balls. For low orders, the generators' matrix is used to define a centrally symmetric convex polytope in (c). We analyze a related application, where we evaluate methods of zonotope approximation in engineering.

volesti: Volume Approximation and Sampling for Convex Polytopes in R

“…For V-and Z-polytopes, the cost per step of BiW is comparable with that of CDHR. Moreover, it converges fast to the uniform distribution (Chalkis et al, 2019). The fact that all above walks are implemented in volesti enable us to empirically evaluate their mixing time using R (e.g., Figure 3).…”

“…For example, the asymptotic analysis by hides some large constants in the complexity, and in Lee and Vempala (2018), the step of the random walk used for sampling is too small to be an efficient choice in practice. Therefore, practical algorithms have been designed by relaxing the theoretical guarantees and applying new algorithmic and statistical techniques to perform efficiently while at the same time meeting the requirements for high accuracy results (Emiris and Fisikopoulos, 2014;Cousins and Vempala, 2016;Chalkis et al, 2019).…”

“…In this paper, we present volesti (Fisikopoulos et al, 2020), an R package containing a variety of high-dimensional MCMC methods for sampling from multivariate distributions restricted to a convex polytope and randomized algorithms for volume estimation of convex polytopes. In particular, it includes efficient implementations of three practical volume algorithms-Sequence of Balls (SoB) (Emiris and Fisikopoulos, 2014), Cooling Gaussians (CG) (Cousins and Vempala, 2016), and Cooling convex Bodies (CB) (Chalkis et al, 2019). In addition to volume estimation, volesti provides efficient implementations for Random-Directions and Coordinate-Directions Hit and Run (RDHR and CDHR) (Smith, 1984), Ball Walk (BaW) (Hastings, 1970), Billiard Walk (BiW) (Polyak and Gryazina, 2014).…”

Sampling the feasible sets of SDPs and volume approximation

“…We employ the annealing schedule from [11] to minimize k, without computing neither an enclosed r B n nor an enclosing ball RB n of S. We do so by probabilistically bounding each ratio of Eq. ( 9) in…”

Geometric algorithms for sampling the flux space of metabolic networks

“…For example, HR mixes after O(d 3 ) [29] steps for well rounded convex bodies and CDHR after a polynomial, in the diameter and the dimension, number of steps [27,34]. However, extended practical results have shown that both CDHR and HR converges after O(d 2 ) steps [9,12,18]. Billiard Walk (BW) [16] is a random walk that employs linear trajectories in a convex body with boundary reflections.…”