Journal of Global Optimization

, Volume 43, Issue 2–3, pp 329–356 | Cite as

The interacting-particle algorithm with dynamic heating and cooling

Article

Abstract

We consider an interacting-particle algorithm which is population-based like genetic algorithms and also has a temperature parameter analogous to simulated annealing. The temperature parameter of the interacting-particle algorithm has to cool down to zero in order to achieve convergence towards global optima. The way this temperature parameter is tuned affects the performance of the search process and we implement a meta-control methodology that adapts the temperature to the observed state of the samplings. The main idea is to solve an optimal control problem where the heating/cooling rate of the temperature parameter is the control variable. The criterion of the optimal control problem consists of user defined performance measures for the probability density function of the particles’ locations including expected objective function value of the particles and the spread of the particles’ locations. Our numerical results indicate that with this control methodology the temperature fluctuates (both heating and cooling) during the progress of the algorithm to meet our performance measures. In addition our numerical comparison of the meta-control methodology with classical cooling schedules demonstrate the benefits in employing the meta-control methodology.

Keywords

Interacting-particle algorithm Meta-control Optimal control Global optimization Simulated annealing Cooling schedule 

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References

  1. Ali M.M., Khompatraporn C. and Zabinsky Z.B. (2005). A numerical evaluation of several stochastic algorithms on selected continuous global optimization test problems. J. Glob. Optim. 31(4): 631–672 CrossRefGoogle Scholar
  2. (2000). Nonlinear model predictive control. Progress in Systems and Control Theory, vol. 26. Birkhäuser Verlag, Basel Google Scholar
  3. Azizi N. and Zolfaghari S. (2004). Adaptive temperature control for simulated annealing: a comparative study. Comput. Oper. Res. 31: 2439–2451 CrossRefGoogle Scholar
  4. Bertsekas D.P. (1995). Dynamic Programming and Optimal Control, vol. 1. Athena Scientific, Belmont Google Scholar
  5. Cerf R. (1996). A new genetic algorithm. Ann. Appl. Probab. 6(3): 778–817 CrossRefGoogle Scholar
  6. Dunkl C.F. and Xu Y. (2001). Orthogonal Polynomials of Several Variables. Cambridge University Press, Cambridge Google Scholar
  7. Ingber L. (1996). Adaptive simulated annealing (asa): lessons learned. Control Cybern. 25: 22–54 Google Scholar
  8. Kirkpatrick S., Gelatt C.D.J. and Vecchi M.P. (1983). Optimisation by simulated annealing. Science 220: 671–680 CrossRefGoogle Scholar
  9. Kohn W., Zabinsky Z.B. and Brayman V. (2006). Optimization of algorithmic parameters using a meta-control approach. J. Glob. Optim. 34(2): 293–316 CrossRefGoogle Scholar
  10. Kolonko M. and Tran M.T. (1997). Convergence of simulated annealing with feedback temperature schedules. Probab. Eng. Inform. Sci. 11: 279–304 CrossRefGoogle Scholar
  11. Lovász L. (1999). Hit-and-run mixes fast. Math. Program. 86: 443–461 CrossRefGoogle Scholar
  12. Mitter S.K. (1966). Successive approximation methods for the solution of optimal control problems. Automatica 3: 135–149 CrossRefGoogle Scholar
  13. Molvalioglu, O., Zabinsky, Z.B., Kohn, W.: Multi-particle simulated annealing. In: Törn, A., Zilinskas, J.(eds.) Models and Algorithms for Global Optimization. ptimization and its applications, vol. 4. Springer (2006)Google Scholar
  14. Molvalioglu, O., Zabinsky, Z.B., Kohn, W.: Meta-control of an Interacting-particle Algorithm for Global Optimization. Technical report, University of Washington (2007)Google Scholar
  15. Moral P.D. (2004). Feynman-Kac Formulae: Genological and Interacting Particle Systems with Applications. Springer-Verlag, New York Google Scholar
  16. Moral P.D. and Miclo L. (2006). Dynamiques recuites de type Feynman-Kac: résultats précis et conjectures (French). ESAIM: Probab. Stat. 10: 76–140 CrossRefGoogle Scholar
  17. Munakata T. and Nakamura Y. (2001). Temperature control for simulated annealing. Phys. Rev. E 64(4): 46–127 CrossRefGoogle Scholar
  18. Scott D. (1992). Multivariate Density Estimation: Theory, Practice and Visualization. Wiley, New York Google Scholar
  19. Sharpe F.W. (1994). The sharpe ratio. J. Portfolio Manage. 21: 49–59 CrossRefGoogle Scholar
  20. Shen Y., Kiatsupaibul S., Zabinsky Z.B. and Smith R.L. (2007). An analytically derived cooling schedule for simulated annealing. J. Glob. Optim. 38: 333–365 CrossRefGoogle Scholar
  21. Smith R.L. (1984). Efficient Monte Carlo procedures for generating points uniformly distributed over bounded region. Oper. Res. 32: 1296–1308 CrossRefGoogle Scholar
  22. Srinivas M. and Patnaik L.M. (1994). Genetic algorithms: a survey. IEEE Comp. 27(6): 17–26 Google Scholar
  23. Triki E., Collette Y. and Siarry P. (2005). A theoretical study on the behavior of simulated annealing leading to a new cooling schedule. Eur. J. Oper. Res. 166(1): 77–92 CrossRefGoogle Scholar
  24. Zabinsky Z.B. (2003). Stochastic Adaptive Search for Global Optimization. Kluwer Academic Publishers, Boston Google Scholar

Copyright information

© Springer Science+Business Media, LLC. 2008

Authors and Affiliations

  • Orcun Molvalioglu
    • 1
  • Zelda B. Zabinsky
    • 1
  • Wolf Kohn
    • 1
  1. 1.Department of Industrial EngineeringUniversity of WashingtonSeattleUSA

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