Keywords: particle swarm, coefficients, neighbourhood topology, convergence.

Individually, there are three basic factors that govern a particle's trajectory: 1) the inertia from its previous displacement; 2) the attraction to its own best experience; and 3) the attraction to a given neighbour's best experience. The importance awarded to each factor is regulated by three coefficients: 1) the inertia; 2) the individuality; and 3) the sociality weights. The other important question regarding the particles' behaviour is how to define the social attractor in the velocity equation, which governs the social behaviour. This leads to the design of different neighbourhood topologies within the swarm, where the lower the number of interconnections the slower the convergence. An extensive study of neighbourhood topologies can be found in [1]. There is always the need for a trade-off between the explorative and the exploitative behaviour. The former is more reluctant to get trapped in sub-optimal solutions whereas the latter is better for a fine-grain search. This trade-off may be controlled by both the coefficients' settings and the neighbourhood topology.

The aim of this chapter is two-fold: first to offer some guidelines on the impact of different coefficients' settings on the speed and form of convergence; and second to illustrate their combined effect on the neighbourhood topology. Thus, the conver-gence region of the plane 'inertia weight (w)–acceleration coefficient (phi)' is pre-sented, and the effect on the trajectory of a deterministic and isolated particle is ana-lyzed for different sub-regions. Related studies can be found in [2], and [3]. The ef-fect of setting the individuality (iw) and sociality weights (sw) to different values for a given acceleration weight (aw) is also explored. Experiments are performed for a small swarm and a one-dimensional problem to analyze the trajectories and observe whether the conclusions derived from the study of the deterministic particle hold for the full algorithm. Finally, experiments on two 30-dimensional problems are per-formed for different combinations between two sets of coefficients' settings and three neighbourhood topologies. The results and convergence curves illustrate the effect that the coefficients, the neighbourhood topologies, and their different combi-nations have on the performance of the optimizer. Thus the user can decide upon the coefficients and the neighbourhoods according to the type of search desired.

[1]

R. Mendes, "Population Topologies and Their Influence in Particle Swarm Performance", Ph.D. Thesis, Escola de Engenharia, Universidade do Minho, Portugal, 2004.

[2]

E. Ozcan, C. K. Mohan, "Analysis of a simple particle swarm optimization system", Intelligent Engineering Systems Through Artificial Neural Networks: Volume 8, 253-258, ASME books, 1998.

[3]

M. Clerc, J. Kennedy, "The Particle Swarm-Explosion, Stability, and Convergence in a Multidimensional Complex Space", IEEE Transactions on Evolutionary Computation, 6(1), 58-73, 2002. doi:10.1109/4235.985692

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