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A theoretical framework for simulated annealing

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Abstract

Simulated Annealing has been a very successful general algorithm for the solution of large, complex combinatorial optimization problems. Since its introduction, several applications in different fields of engineering, such as integrated circuit placement, optimal encoding, resource allocation, logic synthesis, have been developed. In parallel, theoretical studies have been focusing on the reasons for the excellent behavior of the algorithm. This paper reviews most of the important results on the theory of Simulated Annealing, placing them in a unified framework. New results are reported as well.

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References

  1. E. H. L. Aarts and P. J. M. van Laarhoven. Statistical Cooling: A General Approach to Combinatorial Optimization Problems.Philips J. Res.,40:193–226, 1985.

    MathSciNet  Google Scholar 

  2. E. H. L. Aarts and P. J. M. van Laarhoven.Simulated Annealing: Theory and Applications. Reidel, Dordrecht, 1987.

    MATH  Google Scholar 

  3. S. Anily and A. Federgruen. Probabilistic Analysis of Simulated Annealing Methods. Technical Report, Graduate School of Business, Columbia University, New York, 1985.

    Google Scholar 

  4. S. Anily and A. Federgruen. Simulated Annealing Methods with General Acceptance Probabilities.J. Appl. Probab.,24: 657–667, 1987.

    Article  MATH  MathSciNet  Google Scholar 

  5. C. R. Aragon, D. S. Johnson, L. A. McGeoch, and C. Schevon. Simulated Annealing Performance Studies. Workshop on Statistical Physics in Engineering and Biology, April, 1984.

  6. P. Banerjee and M. Jones. A Parallel Simulated Annealing Algorithm for Standard Cell Placement on a Hypercube Computer.Proc. ICCAD, pages 34–37, 1986.

  7. K. Binder.Monte Carlo Methods in Statistical Physics. Springer Verlag, Berlin, 1978.

    Google Scholar 

  8. I. O. Bohachevsky, M. E. Johnson, and M. L. Stein. Generalized Simulated Annealing for Function Optimization.Technometrics,28 (3): 209–217, August, 1986.

    Article  MATH  Google Scholar 

  9. A. Casotto, F. Romeo, and A. Sangiovanni-Vincentelli. A Parallel Simulated Annealing Algorithm for the Placement of Macro-Cells.IEEE Trans. Computer-Aided Design,6: 838–847, September, 1987.

    Article  Google Scholar 

  10. A. Casotto and A. Sangiovanni-Vincentelli. Placement of Standard Cells Using Simulated Annealing on the Connection Machine.Proc. ICCAD, pages 350–353, 1987.

  11. D. P. Connors and P. R. Kumar. Balance of Recurrence Orders in Time-Inhomogeneous Markov Chains with Applications to Simulated Annealing.Probab. Engrg. Inform. Sci.,2 (2): 157–184, 1988.

    MATH  Google Scholar 

  12. F. Darema-Rogers, S. Kirkpatrick, and V. A. Norton. Simulated Annealing on Shared Memory Parallel Systems.IBM J. Res. Develop., 1987.

  13. W. B. Davenport.Probability and Random Processes: An Introduction for Applied Scientists and Engineers. McGraw-Hill, New York, 1970.

    Google Scholar 

  14. U. Faigle and R. Schrader. On the Convergence of Stationary Distributions in Simulated Annealing Algorithms.Inform. Process. Lett.,27 (4): 189–194, April, 1988.

    Article  MATH  MathSciNet  Google Scholar 

  15. D. Freedman.Markov Chains. Holden-Day, San Francisco, CA, 1971.

    MATH  Google Scholar 

  16. S. Gelfand. Analysis of Simulated Annealing Type of Algorithms. Technical Report LIDS-TH-1688, Massachusetts Institute of Technology, Cambridge, MA., 1987.

    Google Scholar 

  17. S. Gelfand and S. Mitter. Analysis of Simulated Annealing for Optimization.Proc. 24th Conf. on Decision and Control, pages 779–786, December 1985.

  18. S. Gelfand and S. Mitter. Simulated Annealing with Noisy or Imprecise Energy Measurements. Technical Report, Massachusetts Institute of Technology, Cambridge, MA, 1987.

    Google Scholar 

  19. S. Geman and D. Geman. Stochastic Relaxation, Gibbs Distributions, and the Bayesian Restoration of Images.IEEE Trans. Pattern Anal. Mach. Intell.,6: 721–741, 1984.

    MATH  Google Scholar 

  20. B. Gidas. Non-Stationary Markov Chains and Convergence of the Annealing Algorithm.J. Statist. Phys.,39: 73–131, 1984.

    Article  MathSciNet  Google Scholar 

  21. B. V. Gnedenko and A. Y. Khinchin.An Elementary Introduction to the Theory of Probability, Freeman, San Francisco, CA, 1961.

    MATH  Google Scholar 

  22. B. Hajek. Cooling Schedules for Optimal Annealing.Math. Oper. Res.,13 (2): 311–329, 1985.

    Article  MathSciNet  Google Scholar 

  23. J. M. Hammersley and D. C. Handscomb.Monte Carlo Methods, Wiley, New York, 1964.

    MATH  Google Scholar 

  24. W. K. Hastings. Monte Carlo Sampling Methods Using Markov Chains and Their Applications.Biometrika,57 (1): 97–109, 1970.

    Article  MATH  Google Scholar 

  25. D. Hillis.The Connection Machine. MIT Press, Cambridge, MA, 1985.

    Google Scholar 

  26. P. G. Hoel.Introduction to Mathematical Statistics. Wiley, New York, 1962.

    Google Scholar 

  27. R. Holley and D. Stroock. Simulated Annealing via Sobolev Inequalities.Comm. Math. Phys.,115 (4): 553–569, April, 1988.

    Article  MATH  MathSciNet  Google Scholar 

  28. M. D. Huang, F. Romeo, and A. Sangiovanni-Vincentelli. An Efficient General Cooling Schedule for Simulated Annealing,Proc. ICCAD, pages 381–384, 1986.

  29. S. Hustin. Tim, a New Standard Cell Placement Program Based on the Simulated Annealing Algorithm. Master's thesis, University of California, Berkeley, CA, March, 1988.

    Google Scholar 

  30. D. L. Isaacson and R. W. Madsen.Markov Chains: Theory and Applications. Wiley, New York, 1976.

    MATH  Google Scholar 

  31. J. G. Kemeny and J. L. Snell.Finite Markov Chains. Springer-Verlag, New York, 1976.

    MATH  Google Scholar 

  32. S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi. Optimization by Simulated Annealing.Science,220(4598): 671–680, 13 May, 1983.

    Article  MathSciNet  Google Scholar 

  33. S. A. Kravitz and R. A. Rutenbar. Multiprocessor-based Placement by Simulated Annealing.Proc. 23rd Design Automation Conf., pages 567–573, 1986.

  34. J. Lam and J-M. Delosme. Logic Minimization Using Simulated Annealing.Proc. ICCAD, pages 348–352, 1986.

  35. J. Lam and J.-M. Delosme. An Adaptive Annealing Schedule. Technical Report 8608, University of Yale, New Haven, C., 1987.

    Google Scholar 

  36. J. Lam and J-M. Delosme. Simulated Annealing: A Fast Heuristic for Some Generic Layout Problems.Proc. ICCAD, pages 510–513, 1988.

  37. M. Lundy and A. Mees. Convergence of the Annealing Algorithm.Math. Programming,34: 111–124, 1986.

    Article  MATH  MathSciNet  Google Scholar 

  38. R. W. Madsen and D. L. Isaacson. Strongly Ergodic Behavior for Non-Stationary Markov Processes.Ann. Prob.,1 (2): 329–335, 1973.

    Article  MATH  MathSciNet  Google Scholar 

  39. N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, and A. H. Teller. Equation of State Calculations by Fast Computer Machines.J. Chem. Phys.,21: 1087, 1953.

    Article  Google Scholar 

  40. D. Mitra, F. Romeo, and A. Sangiovanni-Vincentelli. Convergence and Finite-Time Behavior of Simulated Annealing.Proc. 24th Conf. on Decision and Control, December, 1985.

  41. D. Mitra, F. Romeo, and A. Sangiovanni-Vincentelli. Convergence and Finite-Time Behavior of Simulated Annealing.Adv. Appl. Probab.,18: 747–771, 1986.

    Article  MATH  MathSciNet  Google Scholar 

  42. J. D. Nulton and P. Salamon. Statistical Mechanics of Combinatorial Optimization.Phys. Rev. A,37(4): 1351–1356, February, 1988.

    Article  MathSciNet  Google Scholar 

  43. R. H. J. M. Otten and L. P. P. P. van Ginneken. Floorplan Design Using Simulated Annealing.Proc. ICCAD, pages 96–98, 1984.

  44. R. H. Otten and L. P. van Ginneken.Simulated Annealing: The Algorithm. Kluwer, Boston, MA, 1989.

    Google Scholar 

  45. P. H. Peskun. Optimum Monte Carlo Sampling Using Markov Chains.Biometrika,60(3): 607–612, 1973.

    Article  MATH  MathSciNet  Google Scholar 

  46. B. D. Ripley.Stochastic Simulation. Wiley, New York, 1987.

    MATH  Google Scholar 

  47. F. Romeo. Simulated Annealing: Theory and Applications to Layout Problems. Ph. D. thesis, University of California, Berkeley, CA, May 1989.

    Google Scholar 

  48. Y. Rossier, M. Troyon, and T. Liebling. Probabilistic Exchange Algorithms and Euclidean Traveling Salesman Problems. Technical Report RO 851125, EPF, Lousanne, 1985.

    Google Scholar 

  49. W. Rudin.Principles of Mathematical Analysis. McGraw-Hill, New York, 1976.

    MATH  Google Scholar 

  50. A. Sangiovanni-Vincentelli. Editor's Foreword.Algorithmica, this issue.

  51. E. Seneta.Non-negative Matrices and Markov Chains. Springer-Verlag, New York, 1981.

    MATH  Google Scholar 

  52. G. B. Sorkin. Combinatorial Optimization, Simulated Annealing and Fractals. Master's thesis. University of California, Berkeley, CA, December, 1987.

    Google Scholar 

  53. G. B. Sorkin. Rapidly Mixing Analysis of Simulated Annealing on Fractals.Proc. of Sixth Annual MIT VLSI Conf., pages 331–351, April, 1990.

  54. G. B. Sorkin. Efficient Simulated Annealing on Fractal Energy Landscapes.Algorithmica, this issue, pp. 367–418, 1991.

  55. P. N. Strenski. Optimal Annealing Schedules: A Case Study. Technical Report 12923, I.B.M. T. J. Watson Research Center, Yorktown Heights, NY, 1987.

    Google Scholar 

  56. P. N. Strenski and S. Kirkpatrick. Analysis of Finite Length Annealing Schedules.Algorithmica, this issue, pp. 346–366, 1991.

  57. H. C. Tijms.Stochastic Modeling and Analysis: A Computational Approach, Wiley, Chichester, 1986.

    Google Scholar 

  58. J. N. Tsitsiklis.Markov Chains with Rare Transitions and Simulated Annealing. MIT Laboratory for Information and Decision Systems, Cambridge, MA, August, 1985.

    Google Scholar 

  59. V. Černy. Thermodynamical Approach to the Traveling Salesman Problem: An Efficient Simulation Algorithm.J. Optim. Theory Appl.,45: 41–51, 1985.

    Article  MATH  MathSciNet  Google Scholar 

  60. S. White, Concept of Scale in Simulated Annealing.Proc. ICCD, pages 646–651, 1984.

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Authors and Affiliations

  1. Department of Electrical Engineering and Computer Science, University of California, 94720, Berkeley, CA, USA

    Fabio Romeo & Alberto Sangiovanni-Vincentelli

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  1. Fabio Romeo
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  2. Alberto Sangiovanni-Vincentelli
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Additional information

Communicated by Alberto Sangiovanni-Vincentelli.

This research was sponsored by SRC and DARPA monitored by SNWSC under contract numbers N00039-87-C-012 and N00039-88-C-0292.

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Romeo, F., Sangiovanni-Vincentelli, A. A theoretical framework for simulated annealing. Algorithmica 6, 302–345 (1991). https://doi.org/10.1007/BF01759049

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  • DOI: https://doi.org/10.1007/BF01759049

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