Iterant recombination with one-norm minimization for multilevel Markov chain algorithms via the ellipsoid method
- 59 Downloads
Recently, it was shown how the convergence of a class of multigrid methods for computing the stationary distribution of sparse, irreducible Markov chains can be accelerated by the addition of an outer iteration based on iterant recombination. The acceleration was performed by selecting a linear combination of previous fine-level iterates with probability constraints to minimize the two-norm of the residual using a quadratic programming method. In this paper we investigate the alternative of minimizing the one-norm of the residual. This gives rise to a nonlinear convex program which must be solved at each acceleration step. To solve this minimization problem we propose to use a deep-cuts ellipsoid method for nonlinear convex programs. The main purpose of this paper is to investigate whether an iterant recombination approach can be obtained in this way that is competitive in terms of execution time and robustness. We derive formulas for subgradients of the one-norm objective function and the constraint functions, and show how an initial ellipsoid can be constructed that is guaranteed to contain the exact solution and give conditions for its existence. We also investigate using the ellipsoid method to minimize the two-norm. Numerical tests show that the one-norm and two-norm acceleration procedures yield a similar reduction in the number of multigrid cycles. The tests also indicate that one-norm ellipsoid acceleration is competitive with two-norm quadratic programming acceleration in terms of running time with improved robustness.
KeywordsMarkov chain Iterant recombination Ellipsoid algorithm Multigrid Convex programming
Unable to display preview. Download preview PDF.
- 3.Berman A., Plemmons R.J.: Nonnegative Matrices in the Mathematical Sciences. SIAM, Philadelphia, PA (1987)Google Scholar
- 4.Bertsimas D., Tsitisklis J.N.: Introduction to Linear Optimization. Athena Scientific, Belmont, MA (1997)Google Scholar
- 23.Horton, G., Leutenegger, S.T.: A multi-level solution algorithm for steady-state Markov chains. In: Proceedings of the 1994 ACM SIGMETRICS Conference on Measurement and Modeling of Computer Systems, pp. 191–200 (1994)Google Scholar
- 24.Iudin D.B., Nemirovskii A.S.: Informational complexity and effective methods of solution for convex extremal problems. Matekon Transl. Russ. East Eur. Math. Econ. 13, 3–25 (1976)Google Scholar
- 25.Khachiyan L.G.: A polynomial algorithm in linear programming. Sov. Math. Doklady 20, 191–194 (1976)Google Scholar
- 27.Krieger U.R.: Numerical solution of large finite Markov chains by algebraic multigrid techniques. In: Stewart, W. (eds) Numerical Solution of Markov Chains, pp. 403–424. Kluwer, Dordrecht (1995)Google Scholar
- 29.Leutenegger, S.T., Horton, G.: On the utility of the multi-level algorithm for the solution of nearly completely decomposable Markov chains. Tech. Rep. 94-44, ICASE (1994)Google Scholar
- 37.Shor N.Z.: Cut-off method with space extension in convex programming problems. Cybernetics 13, 94–96 (1977)Google Scholar
- 39.Stewart W.J.: An Introduction to the Numerical Solution of Markov Chains. Princeton University Press, Princeton, NJ (1994)Google Scholar
- 40.Takahashi, Y.: A lumping method for numerical calculations of stationary distributions of Markov chains. Tech. Rep. B-18, Department of Information Sciences, Tokyo Institute of Technology (1975)Google Scholar
- 41.Treister E., Yavneh I.: On-the-fly adaptive smoothed aggregation for Markov chains. SISC 33, 2927–2949 (2011)Google Scholar