Journal of Theoretical Probability

, Volume 5, Issue 2, pp 223–249 | Cite as

Singular perturbed Markov chains and exact behaviors of simulated annealing processes

  • Chii-Ruey Hwang
  • Shuenn-Jyi Sheu


We study asymptotic properties of discrete and continuous time generalized simulated annealing processesX(·) by considering a class of singular perturbed Markov chains which are closely related to the large deviation of perturbed diffusion processes. Convergence ofX(t) in probability to a setS0 of desired states, e.g., the set of global minima, and in distribution to a probability concentrated onS0 are studied. The corresponding two critical constants denoted byd and Λ withd≤Λ are given explicitly. When the cooling schedule is of the formc/logt, X(t) converges weakly forc>0. Whether the weak limit depends onX(0) or concentrates onS0 is determined by the relation betweenc, d, and Λ. Whenc>Λ, the expression for the rate of convergence for each state is also derived.

Key Words

Simulated annealing singular perturbed Markov chains large deviation 


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  1. 1.
    Cerny, V. (1982).A Thermodynamical Approach to the Traveling Salesman Problem: An Efficient Simulation Algorithm, Institute of Physics and Biophysics, Comenius University, Bratislava, Czechoslovakia.Google Scholar
  2. 2.
    Chiang, T.-S., and Chow, Y. (1988). On the convergence rate of annealing processes.SIAM J. Control Opt. 17, 1455–1470.Google Scholar
  3. 3.
    Chiang, T.-S., and Chow, Y. (1989). A limit theorem for a class of inhomogeneous Markov processes.Ann. Prob. 17, 1483–1502.Google Scholar
  4. 4.
    Chiang, T.-S., Hwang, C.-R., and Sheu, S.-J. (1987). Diffusion for global optimization in ℝn.SIAM J. Control Opt. 25, 737–753.Google Scholar
  5. 5.
    Freidlin, M. I., and Wentzell, A. D. (1984).Random Perturbations of Dynamical Systems, Springer-Verlag, New York.Google Scholar
  6. 6.
    Geman, S., and Geman, D. (1984). Stochastic relaxation, Gibbs distribution, and the Baysian restoration of images.IEEE Trans. Pattern Anal. Mach. Intel. 6, 721–741.Google Scholar
  7. 7.
    Geman, S., and Hwang, C.-R. (1986). Diffusion for global optimization.SIAM J. Control Opt. 24, 1031–1043.Google Scholar
  8. 8.
    Gihman, I. I., and Shorohod, A. V. (1975).The Theory of Stochastic Processes II. Springer-Verlag, New York.Google Scholar
  9. 9.
    Hajek, B. (1985). Cooling schedules for optimal annealing.Math. Oper. Res. 13, 311–329.Google Scholar
  10. 10.
    Hwang, C.-R., and Sheu, S.-J. (1989). On the weak reversibility condition in simulated annealing.Soochow J. Math. 15, 159–170.Google Scholar
  11. 11.
    Hwang, C.-R., and Sheu, S.-J. (1990). Large-time behavior of perturbed diffusion Markov processes with applications to the second eigenvalue problem for Fokker-Planck operators and simulated annealing.Acta Appl. Math. 19, 253–295.Google Scholar
  12. 12.
    Kirkpatrick, S., Gelatt, C. D., and Veochi, M. P. (1983). Optimization by simulated annealing.Science 220, 671–680.Google Scholar
  13. 13.
    Tsitsiklis, J. (1985).Markov Chains with Rare Transitions and Simulated Annealing, Laboratory for Information and Decision Systems, Massachusetts Institute of Technology, Cambridge, MA.Google Scholar
  14. 14.
    van Laarhoven, P. J. M., and Aarts, E. M. L. (1987).Simulated Annealing: Theory and Applications, D. Reidel, Dordrecht.Google Scholar

Copyright information

© Plenum Publishing Corporation 1992

Authors and Affiliations

  • Chii-Ruey Hwang
    • 1
  • Shuenn-Jyi Sheu
    • 1
  1. 1.Institute of MathematicsAcademia SinicaTaipeiTaiwan, Republic of China

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