Skip to main content
Log in

Some results characterizing the finite time behaviour of the simulated annealing algorithm

  • Chance As Necessity
  • Published:
Sadhana Aims and scope Submit manuscript

Abstract

TheSimulated Annealing algorithm is a probabilistic search technique for finding the minimum cost state in a set Ω. The algorithm has been successfully used to obtain near-optimal solutions for problems for which no other effective algorithms exist. For example, problems in integrated circuit layout and in finite impulse response (FIR) filter design have been solved using annealing. In most applications, Ω is finite set, and the annealing algorithm may be modelled as a time-inhomogeneous Markov chain on Ω with transition probabilities that are powers of a time varying parameter ε. It has been shown by several researchers that if ε is driven to 0 sufficiently slowly, then the algorithm will eventually find a minimum cost state in Ω with probability 1. In this paper, we will focus on the finite-time behaviour of the annealing algorithm. In particular, we will summarize some results relating the number of steps taken by the algorithm to the quality of the solutions obtained. These results provide qualitative as well as quantitative information about the status of the annealing algorithm after a finite number of steps. This will be illustrated using some examples.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Apostol T M 1976Introduction to analytic number theory (New York: Springer-Verlag)

    MATH  Google Scholar 

  • 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 (Piscataway, NJ: IEEE Press) pp 1296–1297

    Google Scholar 

  • Catoni O 1991a Sharp large deviation estimates for simulated annealing algorithms.Ann. Inst. Henri Poincare 27: 291–383

    MATH  MathSciNet  Google Scholar 

  • Catoni O 1991b Application of sharp large deviation estimates to optimal cooling schedules.Ann. Inst. Henri Poincare 27: 463–518

    MATH  MathSciNet  Google Scholar 

  • Catoni O 1992 Rough large deviation estimates for simulated annealing algorithms: application to exponential schedules.Ann. Probab. 20: 1109–1140

    Article  MATH  MathSciNet  Google Scholar 

  • Cerny V 1985 Thermodynamical approach to the travelling salesman problem: An efficient simulation algorithm.J. Optim. Theory Appl. 45: 41–51

    Article  MATH  MathSciNet  Google Scholar 

  • Cheeger J 1970 A lower bound for the smallest eigenvalue of the Laplacian. InProblems in analysis (ed.) R C Gunning (Princeton, NJ: University Press) pp 195–199

    Google Scholar 

  • Chiang T-S, Chow Y 1984 On eigenvalues and annealing rates.Math. Oper. Res. 9: 508–511

    MathSciNet  Google Scholar 

  • Chiang T-S, Chow Y 1989 A limit theorem for a class of inhomogeneous Markov processes.Ann. Probab. 17: 1483–1502

    Article  MATH  MathSciNet  Google Scholar 

  • Connors D P, Kumar P R 1988 Balance of recurrence order in time-inhomogeneous Markov chains with application to simulated annealing.Probab. Eng. Inf. Sci. 2: 157–184

    Article  MATH  Google Scholar 

  • 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 Champaign, IL

    Google Scholar 

  • Desai M P, Rao V B 1993 On the convergence of reversible Markov chains.SIAM J. Matrix Anal. 14: 950–966

    Article  MATH  MathSciNet  Google Scholar 

  • Desai M P, Rao V B 1997 Finite time behaviour of slowly cooled annealing chains.Probab. Eng. Inf. Sci. 11: 137–176

    MATH  MathSciNet  Google Scholar 

  • Desai M P, Sunil Kumar, Kumar P R 1994 Quasi-statically cooled Markov chains.Probab. Eng. Inf. Sci. 8: 1–19

    Google Scholar 

  • Diaconis P, Stroock D 1991 Geometric bounds for eigenvalues of Markov chains.Ann. Appl. Probab. 1: 36–61

    Article  MATH  MathSciNet  Google Scholar 

  • Dyer M, Frieze A, Kannan R 1989 A random polynomial algorithm for approximating the volume of a convex body.Proceedings of the 21st Symposium on the Theory of Computation, pp 375–381

  • Fleisher H, Giraldi J, Martin D B, Phoenix R L, Tavel M A 1985 Simulated annealing as a tool for logic optimization in a CAD environment.Proceedings of the IEEE International Conference on Computer-Aided Design, Santa Clara, pp 203–205

  • Garey M R, Johnson D S 1979Computers and intractability: A guide to the theory of NP-completeness (New York: W H Freeman)

    MATH  Google Scholar 

  • Geman S, Geman D 1984 Stochastic relaxation, Gibbs distribution, and the Bayesian restoration of images.IEEE Trans. Pattern Anal. Machine Intell. PAMI-6:

  • Gidas B 1985 Non-stationary Markov chains and convergence of the annealing algorithm.J. Stat. Phys. 39: 73–131

    Article  MATH  MathSciNet  Google Scholar 

  • Hajek B 1988 Cooling schedules for optimal annealing.Math. Oper. Res. 13: 311–329

    Article  MATH  MathSciNet  Google Scholar 

  • Kirkpatrick S, Gelatt C D Jr, Vecchi M P 1983 Optimization by simulated annealing.Science 220: 671–680

    Article  MathSciNet  Google Scholar 

  • van Laarhoven P J M, Aarts E H 1987Simulated annealing: Theory and applications (Dordrecht: D Reidel)

    MATH  Google Scholar 

  • Lawler G, Sokal A 1988 Bounds on theL 2 spectrum for Markov chains and Markov processes: A generalization of Cheeger’s inequality.Trans. Am. Math. Soc. 309: 557–580

    Article  MATH  MathSciNet  Google Scholar 

  • Mazza C 1991 Asymptotic first hitting time distribution of annealing processes. Preprint

  • Metropolis N, Rosenbluth A, Rosenbluth M, Teller A, Teller E 1953 Equation of state calculations by fast computing machines.J. Chem. Phy. 21: 87–92

    Article  Google Scholar 

  • Mitra D, Romeo F, Sangiovanni-Vincentelli A L 1986 Convergence and finite-time behavior of simulated annealing.Adv. Appl. Probab. 18: 747–771

    Article  MATH  MathSciNet  Google Scholar 

  • Mohar B 1988 The Laplacian spectrum of graphs.International Graph Theory Conference, Western Michigan University

  • Otten R H J M, Van Ginneken L P P P 1984 Floorplan design using simulated annealing.Proceedings of the International Conference on Computer-Aided Design (Los Alamitos, CA: IEEE Comput. Press) pp 96–98

    Google Scholar 

  • Papadimitrou C H, Stieglitz K 1981Combinatorial optimization: Algorithms and complexity (Englewood Cliffs, NJ: Prentice-Hall)

    Google Scholar 

  • Sasaki G, Hajek B 1988 The time-complexity of maximum matching by simulated annealing.J. Assoc. Comput. Mach. 35: 387–403

    MathSciNet  Google Scholar 

  • Sechen C T, Sangovianni-Vincentelli A L 1984 The Timber-Wolf placement and routing package.Proceedings of the 1984 Custom Integrated Circuit Conference, Rochester

  • Seneta E 1980Nonnegative matrices and Markov chains. (New York: Springer Verlag)

    Google Scholar 

  • Sinclair A, Jerrum M 1989 Approximate counting, uniform generation and rapidly mixing Markov chains.Inf. Comput. 82: 93–133

    Article  MATH  MathSciNet  Google Scholar 

  • Tsitsiklis J N 1989 Markov chains with rare transitions and simulated annealing.Math. Oper. Res. 14: 70–90

    MATH  MathSciNet  Google Scholar 

  • Vecchi M P, Kirkpatrick S 1983 Global wiring by simulated annealing.IEEE Trans. Comput.-Aided Design CAD-2: 215–222

    Article  Google Scholar 

  • Ventcel A D 1972 On the asymptotics of eigenvalues of matrices with elements of order exp(−V ij /2ε 2).Dokl. Akad. Nauk SSSR 202: 65–68

    Google Scholar 

  • White S 1984 Concepts of scale in simulated annealing.Proceedings of the IEEE International Conference on Computer Design (Los Alamitos, CA: IEEE Comput. Press) pp 646–651

    Google Scholar 

  • Wolberg G, Pavlidis T 1985 Restoration of binary images using stochastic relaxation with annealing.Pattern Recogn. Lett. 3: 375–388

    Article  Google Scholar 

  • Wong D F, Liu C L 1986 A new algorithm for floorplan design.Proceedings of the 23rd Design Automation Conference (New York: ACM Press) pp 1–7

    Google Scholar 

  • Yao X, Liu C L 1990 PLA logic minimization by simulated annealing.Integration, VLSI J. 9: 243–257

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Madhav P Desai.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Desai, M.P. Some results characterizing the finite time behaviour of the simulated annealing algorithm. Sadhana 24, 317–337 (1999). https://doi.org/10.1007/BF02823146

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02823146

Keywords

Navigation