Computational results of an O∗(n4) volume algorithm

Recently an O∗(n4) volume algorithm has been presented for convex bodies by Lovasz and Vempala, where n is the number of dimensions of the convex body. Essentially the algorithm is a series of Monte Carlo integrations. In this paper we describe a computer implementation of the volume algorithm, where we improved the computational aspects of the original algorithm by adding variance decreasing modifications: a stratified sampling strategy, double point integration and orthonormalised estimators. Formulas and methodology were developed so that the errors in each phase of the algorithm can be controlled. Some computational results for convex bodies in dimensions ranging from 2 to 10 are presented as well.

[1]  István Deák,et al.  Random Number Generators and Simulation. , 1991 .

[2]  Miklós Simonovits,et al.  Random walks and an O*(n5) volume algorithm for convex bodies , 1997, Random Struct. Algorithms.

[3]  Miklós Simonovits,et al.  On the randomized complexity of volume and diameter , 1992, Proceedings., 33rd Annual Symposium on Foundations of Computer Science.

[4]  László Lovász,et al.  Hit-and-run mixes fast , 1999, Math. Program..

[5]  Miklós Simonovits,et al.  How to compute the volume in high dimension? , 2003, Math. Program..

[6]  S. Vempala,et al.  Hit-and-Run from a Corner , 2006 .

[7]  M. Simonovits,et al.  Random walks and an O * ( n 5 ) volume algorithm for convex bodies , 1997 .

[8]  S. Vempala,et al.  Where to Start a Geometric Random Walk , 2003 .

[9]  M. Rudelson.  Random Vectors in the Isotropic Position , 1996, math/9608208.

[10]  Santosh S. Vempala,et al.  Simulated annealing in convex bodies and an O*(n4) volume algorithm , 2006, J. Comput. Syst. Sci..

[11]  Santosh S. Vempala,et al.  Simulated Annealing for Convex Optimization , 2004 .

[12]  Martin E. Dyer,et al.  A Random Polynomial Time Algorithm for Approximating the Volume of Convex Bodies , 1989, STOC.

[13]  Santosh S. Vempala,et al.  Simulated annealing in convex bodies and an O*(n/sup 4/) volume algorithm , 2003, 44th Annual IEEE Symposium on Foundations of Computer Science, 2003. Proceedings..

[14]  S. Vempala,et al.  The geometry of logconcave functions and sampling algorithms , 2007 .

[15]  Miklós Simonovits,et al.  Random Walks in a Convex Body and an Improved Volume Algorithm , 1993, Random Struct. Algorithms.

[16]  Robert L. Smith.  The hit-and-run sampler: a globally reaching Markov chain sampler for generating arbitrary multivariate distributions , 1996, Winter Simulation Conference.

[17]  Robert L. Smith,et al.  Simulated annealing for constrained global optimization , 1994, J. Glob. Optim..

[18]  N. Metropolis,et al.  Equation of State Calculations by Fast Computing Machines , 1953, Resonance.

[19]  J. Hammersley,et al.  Monte Carlo Methods , 1965 .