A Non-homogeneous Firefly Algorithm and Its Convergence Analysis

The firefly algorithm is a swarm-based search algorithm, in which fireflies cooperate with each other to look for the optimal solution to a given optimization problem in a provided search space. Even though the firefly algorithm has exhibited good performance, researchers have not adequately explained how it works and what effects of its control coefficients in terms of theory. Further, classical variants of the algorithm have unexpected parameter settings and limited update laws, notably the homogeneous rule is necessary to be improved in order to efficiently search the whole space as accurate as possible for the optimal solutions to various problems. This study analyzes the trajectory of a single firefly in both the traditional algorithm and an adaptive variant based on our previous study. Accordingly, these analyses lead to general models of the algorithm ? including a set of boundary conditions for selection of the control parameters, which can guarantee the convergence tendencies of all individuals. The numerical experiments on twelve well-suited benchmark functions show the implementation of the proposed adaptive algorithm, which is derived from the analyses, can enhance the search ability of each individual in looking for the optima.