The Annals of Applied Probability

On the problem of exit from cycles for simulated annealing processes--a backward equation approach

Tzuu-Shuh Chiang and Yunshyong Chow

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Abstract

For a simulated annealing process $X_t$ on S with transition rates $q_{ij}(t) = p_{ij} \exp (-(U(i, j))/T(t))$ where $i, j \epsilon S$ and $T(t) \downarrow 0$ in a suitable way, we study the exit distribution $P_{t,i}(X_{\tau} = a)$ and mean exit time $E_{t,i}(\tau)$ of $X_t$ from a cycle c as $t \to \infty$. A cycle is a particular subset of S whose precise definition will be given in Section 1. Here $\tau$ is the exit time of the process from c containing i and a is an arbitrary state not in c. We consider the differential (backward) equation of $P_{t,i}(X_{\tau} = a)$ and $E_{t,i}(\tau)$ and show that $\lim_{t\to\infty}P_{t,i}(X_{\tau} = a)/\exp (-U(c, a) - T(t))$ and $\lim_{t\to\infty E_{t,i}(\tau)/\exp(V(c)/T(t))$ exist and are independent of $i \epsilon c$. The constant $(U(c, a))$ is usually referred to as the cost from c to a and $V(c), (\leq U(c, a))$ is the minimal cost coming out of c. We also obtain estimates of $|P_{t,i}(X_{\tau} = a) - P_{t,j}(X_{\tau} = a)|$ and $|E_{t,i}(\tau)|$ for $i \not= j$ as $t \to \infty$. As an application, we shall show that similar results hold for the family of Markov processes with transition rates $q_{ij}^{\varepsilon} = p_{ij} \exp(-U(i, j)/\varepsilon)$ where $\varepsilon > 0$ is small.

Article information

Source
Ann. Appl. Probab., Volume 8, Number 3 (1998), 896-916.

Dates
First available in Project Euclid: 9 August 2002

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1028903456

Digital Object Identifier
doi:10.1214/aoap/1028903456

Mathematical Reviews number (MathSciNet)
MR1627807

Zentralblatt MATH identifier
0937.60067

Subjects
Primary: 60J27: Continuous-time Markov processes on discrete state spaces 60J99: None of the above, but in this section
Secondary: 15A18: Eigenvalues, singular values, and eigenvectors 15A51 90B40: Search theory

Keywords
Simulated annealing process backward equations cycles

Citation

Chiang, Tzuu-Shuh; Chow, Yunshyong. On the problem of exit from cycles for simulated annealing processes--a backward equation approach. Ann. Appl. Probab. 8 (1998), no. 3, 896--916. doi:10.1214/aoap/1028903456. https://projecteuclid.org/euclid.aoap/1028903456


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References

  • [1] Catoni, O. (1992). Rough large deviation estimates for simulated annealing-application to exponential schedules. Ann. Probab. 20 1109-1146.
  • [2] Chen, D., Feng, J. and Qian, M. (1995). The metastability of exponentially perturbed Markov chains. Chinese Sci. A 25 590-595.
  • [3] Chiang, T. S. and Chow, Y. (1989). On the asy mptotic behavior of some inhomogeneous Markov processes. Ann. Probab. 17 1483-1502.
  • [4] Chiang, T. S. and Chow, Y. (1994). The asy mptotic behavior of simulated annelaing with absorption. SIAM J. Control Optim. 32 1247-1265.
  • [5] Chow, Y. and Hsieh, J. (1992). On occupation times of annealing processes. Bull. Math. Academia Sinica 20 19-26.
  • [6] Freidlin, M. I. and Wentzell, A. D. (1984). Random Perturbations of Dy namical Sy stems. Springer, New York.
  • [7] Geman, S. and Geman, D. (1984). Stochastic relaxation, Gibbs distribution and the Bayesian restoration of images. IEEE Trans. Pattern Anal. Mach. Intelligence 6 721-741.
  • [8] Gidas, B. (1985). Global optimization via the Langevin equation. In Proceedings 24th IEEE Conference on Decision and Control, Fort Lauderdale, FL 774-778. IEEE, New York.
  • [9] Hajek, B. (1988). Cooling schedules for optimal annealing. Math. Oper. Res. 13 311-329.
  • [10] Hwang, C. R. and Sheu, S. J. (1992). Singular perturbed Markov chains and exact behaviors of simulated annealing process. J. Theoret. Probab. 5 223-249.
  • [11] Kirkpatrick, S., Gelatt, C. and Vecchi, M. (1983). Optimization by simulated annealing. Science 220 671-680.
  • [12] Neves, E. J. and Schonmann, R. H. (1991). Critical droplets and metastability for a Glauber dy namics at very low temperatore. Comm. Math. Phy s. 137 209-230.
  • [13] Olivieri, E. and Scoppola, E. (1996). Markov chains with exponentially small transition probabilities: First exit problem from a general domain II. J. Statist. Phy s. 84 987- 1041.
  • [14] Schonmann, R. H. (1994). Slow droplet-driven relaxation of stochastic Ising models in the vicinity of the phase coexistence region. Comm. Math. Phy s. 161 1-49.
  • [15] Seneta, E. (1981). Nonnegative Matrices and Markov Chains, 2nd ed. Springer, New York.
  • [16] Van Laarhoven, P. J. M. and Aarts, E. H. L. (1987). Simulated Annealing: Theory and Applications. Reidel, Dordrecht.