Particle swarm optimization (PSO) is a well known population-based search algorithm,
originally developed by Kennedy and Eberhart in 1995. The PSO has been utilized
in a variety of application domains, providing a wealth of empirical evidence for its
effectiveness as an optimizer. The PSO itself has undergone many alterations subsequent
to its inception, some of which are fundamental to the PSO's core behavior, others have
been more application specific. The fundamental alterations to the PSO have to a large
extent been a result of theoretical analysis of the PSO's particle's long term trajectory.
The most obvious example, is the need for velocity clamping in the original PSO. While
there were empirical fndings that suggested that each particle's velocity was increasing
at a rapid rate, it was only once a solid theoretical study was performed that the reason
for the velocity explosion was understood. There has been a large amount of theoretical
research done on the PSO, both for the deterministic model, and more recently for the
This thesis presents an extension to the theoretical deterministic PSO model. Under the
extended model, conditions for particle convergence to a point are derived. At present
all theoretical PSO research is done under the stagnation assumption, in some form or
another. The analysis done under the stagnation assumption is one where the personal
best and neighborhood best are assumed to be non-changing. While analysis under the
stagnation assumption is very informative, it could never provide a complete description
of a PSO's behavior. Furthermore, the assumption implicitly removes the notion of
a social network structure from the analysis. The model used in this thesis greatly
weakens the stagnation assumption, by instead assuming that each particle's personal
best and neighborhood best can occupy an arbitrarily large number of unique positions.
Empirical results are presented to support the theoretical fndings.