This paper presents an extension of the state of the art theoretical model utilized for understanding the stability criteria of the particles in particle swarm optimization algorithms. Conditions for order-1 and order-2 stability are derived by modeling, in the simplest case, the expected value and variance of a particle's personal and neighborhood best positions as convergent sequences of random variables. Furthermore, the condition that the expected value and variance of a particle's personal and neighborhood best positions are convergent sequences is shown to be a necessary condition for order-1 and order-2 stability. The theoretical analysis presented is applicable to a large class of particle swarm optimization variants.

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This paper presents an objective function specially designed for the convergence analysis of a number of particle swarm optimization (PSO) variants. It was found that using a specially designed objective function for convergence analysis is both a simple and valid method for performing assumption free convergence analysis. It was also found that the canonical particle swarm's topology did not have an impact on the parameter region needed to ensure convergence. The parameter region needed to ensure convergent particle behavior was empirically obtained for the fully informed PSO, the bare bones PSO, and the standard PSO 2011 algorithm. In the case of the bare bones PSO and the standard PSO 2011, the region needed to ensure convergent particle behavior differs from previous theoretical work. The difference in the obtained regions in the bare bones PSO is a direct result of the previous theoretical work relying on simplifying assumptions, specifically the stagnation assumption. A number of possible causes for the discrepancy in the obtained convergent region for the standard PSO 2011 are given.

This article investigates various aspects of angle modulated particle swarm optimisers (AMPSO). Previous attempts at improving the algorithm have only been able to produce better results in a handful of test cases. With no clear understanding of when and why the algorithm fails, improving the algorithm's performance have proved to be a difficult and sometimes blind undertaking. Therefore, the aim of this study is to identify the circumstances under which the algorithm might fail, and to understand and provide evidence for such cases. It is shown that the general assumption that good solutions are grouped together in the search space does not hold for the standard AMPSO algorithm or any of its existing variants. The problem is explained by specific characteristics of the generating function used in AMPSO. Furthermore, it is shown that the generating function also prevents particle velocities from decreasing, hindering the algorithm's ability to exploit the binary solution space. Methods are proposed to both confirm and potentially solve the problems found in this study. In particular, this study addresses the problem of finding suitable generating functions for the first time. It is shown that the potential of a generating function to solve arbitrary binary optimisation problems can be quantified. It is further shown that a novel generating function with a single coefficient is able to generate solutions to binary optimisation problems with fewer than four dimensions. The use of ensemble generating functions is proposed as a method to solve binary optimisation problems with more than 16 dimensions.