Swarm/flock optimization algorithms as continuous dynamic systems

A new general typology of optimization algorithms, inspired to classical swarm intelligence, is presented. They are obtained by translating the numerical swarm/flock-based algorithms into differential equations in the time domain and employing analytical closed-forms written in the continuum. The use of circulant matrices for the representation of the connections among elements of the flock allowed us to analytically integrate the differential equations by means of a time-windowing approach. The result of this integration provides functions of time that are closed-forms, suitable for describing the trajectories of the flock members: they are directly used to update the position and the velocity of each bird/particle at each step (time window) and consequently they substitute in the continuous algorithm the classical updating rules of the numerical algorithms. Thanks to the closed forms it is also possible to analyze the effects due to the tuning of parameters in terms of exploration or exploitation capabilities. In this way we are able to govern the behavior of the continuous algorithm by means of non stochastic tuning of parameters. The proposed continuous algorithms have been validated on famous benchmark functions, comparing the obtained results with the ones coming from the corresponding numerical algorithms.