Particle Swarm Optimization with Inertia Weight and Constriction Factor

In the original Particle Swarm Optimization (PSO) formulation, convergence of a particle towards its attractors is not guaranteed. A velocity constraint is successful in controlling the explosion, but not in improving the fine-grain search. Clerc and Kennedy studied this system, and proposed constriction methodologies to ensure convergence and to fine tune the search. Thus, they developed different constriction methods according to the correlations among some coefficients incorporated to the system. Type 1” constriction became very popular because the basic update equations remained virtually unmodified, and the original and intuitive metaphor valid. The main drawbacks of this constriction type are that constriction becomes too strong very quickly as the acceleration is increased; that the speed of convergence cannot be easily controlled; and that there is no flexibility to set a desired form of convergence. Another problem is that the method can be found in the literature formulated so as to constrict a PSO system which already includes the inertia weight, for which the calculation of a constriction factor using the formulae provided by Clerc and Kennedy does not guarantee convergence. This paper analyzes Type 1” constriction in detail, for which Type 1 constriction is also relevant. The formulae for Type 1 and Type 1” constriction factors suitable for a PSO algorithm including the inertia weight are provided.