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Balance of Recurrece Order in Time-Inhomogenous Markov Chains with Application to Simulated Annealing

Published online by Cambridge University Press:  27 July 2009

D. P. Connors  and
P. R. Kumar
D. P. Connors
Affiliation:
Department of Electrical and Computer Engineering and the Coordinated Science Laboratory University of Illinois, Urbana, Illinois 61801
P. R. Kumar
Affiliation:
Department of Electrical and Computer Engineering and the Coordinated Science Laboratory University of Illinois, Urbana, Illinois 61801
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Abstract

We define a notion of order of recurrence for the states and transitions of a general class of time-inhomogeneous Markov chains with transition probabilities proportional to powers of a small vanishing parameter. These orders are shown to satisfy a balance equation across every edge cut in the associated graph. The resulting order balance equations allow computation of the orders of recurrence of the states, and thereby the determination of the asymptotic behavior of the Markov chain.

The method of optimization by simulated annealing is a special case of such Markov processes, and can therefore be treated by means of these balance equations. In particular, in this special situation we show that there holds a detailed balance of order of recurrence across every edge in the graph. Moreover, the sum of the order of recurrence of a state and its cost is shown to be a constant in each connected set of recurrent states. By this approach, we determine the necessary and sufficient condition on the “rate of cooling” to guarantee that a minimum of the optimization problem is hit with probability one. Moreover, the rates of convergence of the probabilities can be deduced from the orders of recurrence.

Type
Articles
Information
Probability in the Engineering and Informational Sciences , Volume 2 , Issue 2 , April 1988 , pp. 157 - 184
Copyright
Copyright © Cambridge University Press 1988

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References

Kirkpatrick, S., Gelatt, C. D. Jr, & Vecchi, M. P. (1983). Optimization by simulated annealing. Science 220 (4598):671–680.CrossRefGoogle ScholarPubMed
Geman, S. & Geman, D. (1984). Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images. IEEE Transactions on Pattern Analysis and Machine Intelligence PAMI-6. no. 6:721–741.Google Scholar
Mitra, D., Romeo, F., & Sangiovanni-Vincentelli, A. (1985). Convergence and finite-time behavior of simulated annealing. Preprint, Electronics Research Laboratory, University of California, Berkeley.CrossRefGoogle Scholar
Gidas, B. (1985). Nonstationary Markov chains and convergence of the annealing algorithm. Journal of Statistical Physics 39:73–131.CrossRefGoogle Scholar
Hajek, B. (1986). Cooling schedules for optimal annealing. Preprint. Department of Electrical Engineering and the Coordinated Science Laboratory, University of Illinois.Google Scholar
Tsitsiklis, J. N. (1985). Markov chains with rare transitions and simulated annealing. Preprint, Laboratory for Information and Decision Systems, Massachusetts institute of Technology.CrossRefGoogle Scholar
Hajek, B. (1985). A tutorial survey of theory and applications of simulated annealing. Proceedings of the 24th IEEE Conference on Decision and Control, Vol. 2, pp. 755–760, Ft. Lauderdale.Google Scholar
Gelfand, S. B. & Mitter, S. K. (1985). Analysis of simulated annealing for optimization. Proceedings of the 24th IEEE Conference on Decision and Control, Vol. 2, pp. 779–786, Ft. Lauderdale.Google Scholar
Chung, K. L. (1974). A Course in Probability Theory. New York: Academic Press.Google Scholar
Bondy, J. A. & Murty, U. S. R. (1976). Graph Theory with Applications. New York: North-Holland.CrossRefGoogle Scholar
Kelly, F. (1979). Reversibility and Stochastic Networks. John Wiley.Google Scholar
Delebecque, F. (1983). A reduction process for perturbed Markov chains. SIAM Journal on Applied Mathematics 43:325–350.CrossRefGoogle Scholar