  Particle Swam Optimization is a population-based and gradient-free
optimization method developed by mimicking social behaviour observed in nature.
Its ability to optimize is not specifically implemented but emerges in the
global level from local interactions. In its canonical version, there are three
factors that govern a particle's trajectory: 1) inertia from its previous
displacement; 2) attraction to its best experience; and 3) attraction to a
given neighbour's best experience. The importance given to each of these
factors is regulated by three coefficients: 1) the inertia; 2) the
individuality; and 3) the sociality weights. Their settings rule the trajectory
of the particle when pulled by these two attractors. Different speeds and forms
of convergence of a particle towards its attractor(s) take place for different
settings of the coefficients. A more general formulation is presented aiming
for a better control of the embedded randomness. Guidelines to select the
coefficients' settings to obtain the desired behaviour are offered. The
convergence speed of the algorithm also depends on the speed of spread of
information within the swarm. The latter is governed by the structure of the
neighbourhood, whose study is beyond the scope of this paper. The objective
here is to help understand the core of the PSO paradigm from the bottom up by
offering some insight into the form of the particles' trajectories, and to
provide some guidelines as to how to decide upon the settings of the
coefficients in the particles' velocity update equation in the proposed
formulation to obtain the type of behaviour desired for the problem at hand.
General-purpose settings are also suggested. The relationship between the
proposed formulation and both the classical and constricted PSO formulations
are also provided.
