Closed Forms for the Fully-Connected Continuous Flock of Starlings Optimization Algorithm

A new typology of swarm-based algorithms which employ analytical closed-forms written in the continuum is presented. The continuous algorithms are firstly introduced by making a simple translation of the numerical swarm-based algorithms into differential equations in the time domain (state equations). The integration of these state equations by using a time windowing approach makes available functions of time that are closed-forms suitable for describing the trajectories of the swarm members for a single time-window. The whole trajectory of a swarm member is then obtained by means of the union of all the paths which have been followed by that member. The proposed continuous algorithms have been validated on famous benchmark functions and the obtained results have been compared with those coming from the corresponding numerical algorithms.