Set-based particle swarm optimisation convergence

Set-based particle swarm optimisation is a swarm-based metaheuristic used to solve combinatorial and discrete optimisation problems. Combinatorial and discrete optimisation problems are fundamentally different from real-valued optimisation problems, and present a uniquely challenging problem to solve and as a result many metaheuristics are not suited to solve combinatorial and discrete optimisation problems. Recently published literature has shown set-based particle swarm optimisation to be very adept at solving a wide range of problems including the multi-dimensional knapsack problem, feature selection, portfolio optimisation, polynomial approximation, clustering, and rule induction. Despite these advancements, no comprehensive study exists on the convergence behaviour of the set-based particle swarm optimisation algorithm. This paper performs the first convergence study of the set-based particle swarm optimisation algorithm and outlines the shortcomings of the algorithm in its current form. Through mathematical proofs, empirical experiments, and sensitivity analysis it is found that set-based particle swarm optimisation does not converge, even under ideal conditions.