A superior random number generator for visiting distribution in GSA

Abstract

Tsallis's generalized simulated annealing (GSA), which has been widely used in many fields as a global optimization tool, is composed of three parts: visiting distribution, accepting rule, and cooling schedule. The most complicated of these is visiting distribution. Although Tsallis and Stariolo did provide a heuristic algorithm to generate a random number for the visiting distribution, empirical simulations have shown that it is inappropriate. In this study a more appropriate algorithm, provided by the inspiration of Mantegna, is composed of two steps. The first is the calculation v = x/|y|^1/α, where x and y are two independent normal variates with zero means and standard deviation σ _x and σ _y respectively. The second is a nonlinear transformation w = {[K(q) − 1]exp(−|v/C(q)|) + 1}v. Theoretical arguments are provided for the choice of α, σ _x , σ _y , and K(q); whereas for C(q), an empirical approximation is used instead. The range of q is in the open interval of (1.0, 3.0). It is shown that both old and new algorithms behave like a normal and Cauchy variates, respectively, when q is in the right limit of 1 or equals 2.0, values which match their corresponding visiting distributions exactly. The criteria for comparison are Kolmogorov's statistic and hypothesis testing. Based on the criteria, extensive simulations confirm that the new algorithm is superior to the old when q ∈ (1.0, 2.5]. As for q ∈ (2.5, 3.0), the values are about the same.

Keywords:

• Generalized simulated annealing
• Kolmogorov's statistic
• Two-hump effect
• Empirical distribution function

Acknowledgements

This research was partially supported by the National Science Council of Taiwan (Project No. NSC 89-2213-E-212-040). Simulation runs in part were performed at a Compaq alpha station, model DS20E, 500 MHz with 2 GB memory at the National Center of High-Performance Computing (NCHC), Taiwan. The center's generous support for CPU time is greatly appreciated. The authors also wish to express appreciation to Dr. Cheryl Rutledge for her editorial assistance.