Hostname: page-component-84b7d79bbc-g7rbq Total loading time: 0 Render date: 2024-07-26T00:33:07.276Z Has data issue: false hasContentIssue false

Finite-Time Behavior of Slowly Cooled Annealing Chains *

Published online by Cambridge University Press:  27 July 2009

Madhav P. Desai
Affiliation:
Department of Electrical Engineering, Indian Institute of Technology, Powai, Mumbai 400076, India
Vasant B. Rao
Affiliation:
IBM, E. Fishkill Facility, Building 306, Zip 3A 1 Hopewell Junction, New York 12533

Abstract

We present results on the finite-time behavior of the discrete-time, finite-space version of the simulated annealing algorithm. The asymptotic and finite-time behavior of the annealing algorithm under slow cooling will be shown to depend on the largest eigenvalue of a certain matrix. To illustrate the utility of our results, we study the slowly cooled annealing algorithm applied to the maximum matching problem and demonstrate a polynomial randomized approximation property of the algorithm.

Type
Research Article
Copyright
Copyright © Cambridge University Press 1997

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Aldous, D. (1987). On the Markov chain simulation method for uniform combinatorial distributions and simulated annealing. Probability in the Engineering and Informational Sciences 1: 3346.Google Scholar
2.Alon, N. (1986). Eigenvalues and expanders. Combinatorica 6(2): 8396.Google Scholar
3.Apostol, T.M. (1976). Introduction to analytic number theory. New York: Springer-Verlag.Google Scholar
4.Catthoor, F., DeMan, H. & Vanderwalle, J. (1986). Investigation of finite word length effects in arbitrary digital filters using simulated annealing. Proceedings of the IEEE International Symposium on Circuits and Systems, San Jose, 05, pp. 12961297.Google Scholar
5.Cerny, V. (1985). Thermodynamical approach to the traveling salesman problem: An efficient simulation algorithm. Journal of Optimization Theory and Applications 45: 4151.Google Scholar
6.Cheeger, J. (1970). A lower bound for the smallest eigenvalue of the Laplacian. In Gunning, R.C.(ed.), Problems in analysis. Princeton, NJ: Princeton University Press, pp. 195199.Google Scholar
7.Chiang, T.-S. & Chow, Y. (1984). On eigenvalues and annealing rates. Mathematics of Operations Research 13(3): 508511.Google Scholar
8.Connors, D.P. & Kumar, P.R. (1988). Balance of recurrence order in time-inhomogeneous Markov chains with application to simulated annealing. Probability in the Engineering and Informational Sciences 2: 157184.Google Scholar
9.Desai, M.P. (1991). An eigenvalue-based approach to the finite time behavior of simulated annealing. Ph.D. thesis, Department of Electrical and Computer Engineering, University of Illinois, Urbana.Google Scholar
10.Desai, M.P., Kumar, S. & Kumar, P.R. (1994). Quasi-statically cooled Markov chains. Probability in the Engineering and Informational Sciences 8(1): 119.Google Scholar
11.Desai, M.P. & Rao, V.B. (1993). On the convergence of reversible Markov chains. SIAM Journal of Matrix Analysis 14(4): 950966.Google Scholar
12.Diaconis, P. & Stroock, D. (1991). Geometric bounds for eigenvalues of Markov chains. Annals of Applied Probability 1(1): 3661.Google Scholar
13.Fleisher, H., Giraldi, J., Martin, D.B., Phoenix, R.L. & Tavel, M.A. (1985). Simulated annealing as a tool for logic optimization is a CAD environment. Proceedings of the IEEE International Conference on Computer-Aided Design, Santa Clara, 11, pp. 203205.Google Scholar
14.Garey, M.R. & Johnson, D.S. (1979). Computers and intractability; a guide to the theory of NP-completeness. New York: W.H. Freeman.Google Scholar
15.Gelfand, S.B. & Mitter, S.K. (1985). Analysis of simulated annealing for optimization. Proceedings of the 24th Conference on Decision and Control, Ft. Lauderdale, FL, 12, pp. 779786.Google Scholar
16.Geman, S. & Geman, D. (1984). Stochastic relaxation, Gibbs distribution, and the Bayesian restoration of images. IEEE Transactions on Pattern Analysis and Machine Intelligence PAMI- 6(6): 721741.Google Scholar
17.Gidas, B. (1985). Non-stationary Markov chains and convergence of the annealing algorithm. Journal of Statistical Physics 39: 73131.Google Scholar
18.Hajek, B. (1988). Cooling schedules for optimal annealing. Mathematics of Operations Research 13(2): 311329.Google Scholar
19.Householder, A.S. (1964). The theory of matrices in numerical analysis. New York: Dover.Google Scholar
20.Holley, R. & Stroock, D. (1988). Simulated annealing via Sobolev inequalities. Communications in Mathematical Physics 115: 553569.Google Scholar
21.Kato, T. (1980). Perturbation theory of linear operators. New York: Springer-Verlag.Google Scholar
22.Kirkpatrick, S., Gelatt, C.D. Jr & Vecchi, M.P. (1983). Optimization by simulated annealing. Science 220(4598): 671680.Google Scholar
23.Lawler, G. & Sokal, A. (1988). Bounds on the L2 spectrum for Markov chains and Markov processes: A generalization of Cheeger's inequality. Transactions of the American Mathematical Society 309: 557580.Google Scholar
24.Metropolis, N., Rosenbluth, A., Rosenbluth, M., Teller, A. & Teller, E. (1953). Equation of state calculations by fast computing machines. Journal of Chemical Physics 21: 8792.Google Scholar
25.Mitra, D., Romeo, F. & Sangiovanni-Vincentelli, A.L. (1986). Convergence and finite-time behavior of simulated annealing. Advances in Applied Probability 18: 747771.Google Scholar
26.Ortega, J. & Rheinboldt, W. (1970). Iterative solutions of nonlinear equations in several variables. New York: Academic Press.Google Scholar
27.Sasaki, G. & Hajek, B. (1988). The time-complexity of maximum matching by simulated annealing. Journal of the Association for Computing Machines 35: 387403.Google Scholar
28.Sechen, C.T. & Sangovianni-Vincentelli, A.L. (1984). The Timber-Wolf placement and routing package. Proceedings of the 1984 Custom Integrated Circuit Conference, Rochester, 05Google Scholar
29.Seneta, E. (1980). Nonnegative matrices and Markov chains. New York: Springer-Verlag.Google Scholar
30.Sinclair, A. & Jerrum, M. (1989). Approximate counting, uniform generation and rapidly mixing Markov chains. Information and Computing 82(1): 93133.Google Scholar
31.Tsitsiklis, J.N. (1985). Markov chains with rare transitions and simulated annealing. Technical Report LIDS-P-1497, Laboratory for Information and Decision Systems, Massachusetts Institute of Technology, Cambridge, MA.Google Scholar
32.Valiant, L.G. (1979). The complexity of enumeration and reliability problems. SI AM Journal of Computing 8(3): 4421.Google Scholar
33.Ventcel, A.D. (1972). On the asymptotics of eigenvalues of matrices with elements of order exp(- Vij/2E2). Doklady Akademii Nauk SSSR 202: 6568.Google Scholar