Parameter Tuning of the Firefly Algorithm by Standard Monte Carlo and Quasi-Monte Carlo Methods

TL;DR

This work investigates parameter tuning for the Firefly Algorithm (FA) using standard Monte Carlo (MC) and quasi-Monte Carlo (QMC) methods, comparing their effects on optimization performance across Sphere, Rosenbrock, and a nonconvex spring design problem. MC and QMC tune FA parameters such as $\beta$, $\gamma$, and $\alpha=\alpha_0 \theta^t$, with MC offering $O(1/\sqrt{N})$ convergence and QMC aiming for $O(1/N)$ under suitable conditions. Results show similar optimal fitness values between MC and QMC across problems, indicating FA robustness to the tuning method, though an F-test reveals variance differences for the spring design. The findings suggest no convincing advantage of QMC over MC for FA parameter tuning in these scenarios, and the work motivates broader validation and deeper theoretical analysis of tuning effects on convergence behavior.

Abstract

Almost all optimization algorithms have algorithm-dependent parameters, and the setting of such parameter values can significantly influence the behavior of the algorithm under consideration. Thus, proper parameter tuning should be carried out to ensure that the algorithm used for optimization performs well and is sufficiently robust for solving different types of optimization problems. In this study, the Firefly Algorithm (FA) is used to evaluate the influence of its parameter values on its efficiency. Parameter values are randomly initialized using both the standard Monte Carlo method and the Quasi Monte-Carlo method. The values are then used for tuning the FA. Two benchmark functions and a spring design problem are used to test the robustness of the tuned FA. From the preliminary findings, it can be deduced that both the Monte Carlo method and Quasi-Monte Carlo method produce similar results in terms of optimal fitness values. Numerical experiments using the two different methods on both benchmark functions and the spring design problem showed no major variations in the final fitness values, irrespective of the different sample values selected during the simulations. This insensitivity indicates the robustness of the FA.