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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
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
(2000). Nonlinear model predictive control. Progress in Systems and Control Theory, vol. 26. Birkhäuser Verlag, Basel
Azizi N. and Zolfaghari S. (2004). Adaptive temperature control for simulated annealing: a comparative study. Comput. Oper. Res. 31: 2439–2451
Bertsekas D.P. (1995). Dynamic Programming and Optimal Control, vol. 1. Athena Scientific, Belmont
Cerf R. (1996). A new genetic algorithm. Ann. Appl. Probab. 6(3): 778–817
Dunkl C.F. and Xu Y. (2001). Orthogonal Polynomials of Several Variables. Cambridge University Press, Cambridge
Ingber L. (1996). Adaptive simulated annealing (asa): lessons learned. Control Cybern. 25: 22–54
Kirkpatrick S., Gelatt C.D.J. and Vecchi M.P. (1983). Optimisation by simulated annealing. Science 220: 671–680
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
Kolonko M. and Tran M.T. (1997). Convergence of simulated annealing with feedback temperature schedules. Probab. Eng. Inform. Sci. 11: 279–304
Lovász L. (1999). Hit-and-run mixes fast. Math. Program. 86: 443–461
Mitter S.K. (1966). Successive approximation methods for the solution of optimal control problems. Automatica 3: 135–149
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)
Molvalioglu, O., Zabinsky, Z.B., Kohn, W.: Meta-control of an Interacting-particle Algorithm for Global Optimization. Technical report, University of Washington (2007)
Moral P.D. (2004). Feynman-Kac Formulae: Genological and Interacting Particle Systems with Applications. Springer-Verlag, New York
Moral P.D. and Miclo L. (2006). Dynamiques recuites de type Feynman-Kac: résultats précis et conjectures (French). ESAIM: Probab. Stat. 10: 76–140
Munakata T. and Nakamura Y. (2001). Temperature control for simulated annealing. Phys. Rev. E 64(4): 46–127
Scott D. (1992). Multivariate Density Estimation: Theory, Practice and Visualization. Wiley, New York
Sharpe F.W. (1994). The sharpe ratio. J. Portfolio Manage. 21: 49–59
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
Smith R.L. (1984). Efficient Monte Carlo procedures for generating points uniformly distributed over bounded region. Oper. Res. 32: 1296–1308
Srinivas M. and Patnaik L.M. (1994). Genetic algorithms: a survey. IEEE Comp. 27(6): 17–26
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
Zabinsky Z.B. (2003). Stochastic Adaptive Search for Global Optimization. Kluwer Academic Publishers, Boston
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Molvalioglu, O., Zabinsky, Z.B. & Kohn, W. The interacting-particle algorithm with dynamic heating and cooling. J Glob Optim 43, 329–356 (2009). https://doi.org/10.1007/s10898-008-9292-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10898-008-9292-z