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Top1. Introduction
Particle swarm optimisation (PSO) is a population based optimisation technique developed by Kennedy and Eberhard (Kennedy & Eberhart, 1995). The particle swarms of optimisation algorithms, in distinction to the swarms in models of animal flocking, swarming and herding, communicate via a social information network rather than on rules dependent on spatial proximity. Each particle remembers its best achieved position, as determined by an objective function, and shares this information with social neighbours who use this knowledge to inform their own exploration.
There have been many attempts to understand the behaviour of the particles in PSO; the studies mainly concentrating on single particle trajectories and formal convergence to a stable point in the absence of particular-particle interaction (Clerc & Kennedy, 2002; Trelea, 2003; Yang &Kamel, 2003; Engelbrecht, 2005; van den Bergh & Engelbrecht, 2006; Blackwell & Bratton, 2007; Poli, 2009; van den Bergh & Engelbrecht, 2010). However an understanding of the dynamics of the interacting particles is elusive. This is due to the complexity of the stochastic particle update (Kennedy, 2003) and the relationship between particle memory, network topology and objective function.
In 2003, Kennedy (Kennedy, 2003) advanced a model of PSO dynamics (‘bare bones’) where the velocity and position update is replaced by Gaussian sampling around the average of the neighbourhood and personal best positions. This provided an arguably simpler model of particle motion.
The original bare bones formulation is not competitive to standard PSO (Richer & Blackwell, 2006; Pena, 2008), but the original idea has been extended. A version based on a broader distribution tail has been proposed (Richer & Blackwell, 2006) with the aim of improving exploration. Krohling considered a version with particle re-initialisation in which all components of particle position are randomised within the search space if a particle has not improved itself over a given number of iterations (Krohling, 2005). A bare bones with component-wise jumps and two social neighbourhoods, BBJ, has also been advanced (Blackwell, 2012). The jumps of BBJ are applied probabilistically and independently of particle performance. The effect is to broaden distribution tails and promote escape from local optima (Blackwell, 2012).
The consequence of these studies is that bare bones swarms can be regarded as effective optimisers in their own right, and are not merely as a model of PSO dynamics.
BBJ, has, by virtue of its dual neighbourhoods and choice of jumping mechanism, a rich set of possible formulations. The particular instances studied here will be termed social (sBBJ) and cognitive BBJ (cBBJ).
Current particle position x in sBBJ does not influence search unless this position coincides with the historical best position achieved. In a sense, one could say that the search is governed by social rather than personal information. The search is focused on the best position, g, of any neighbour within a particular neighbourhood (the μ-neighbourhood) and the search spread is determined by the separation of neighbour best positions in a second social neighbourhood (the σ- neighbourhood). The two neighbourhoods coincide in other bare bones formulations. Good results are obtained by taking a global (the entire swarm) μ-neighbourhood and a small local σ-neighbourhood (Blackwell, 2012). Figuratively speaking, a particle attempts to better itself by copying a public leader (global μ-neighbourhood), yet it also distinguishes itself by imitating the observed degree of non-conformity within a more intimate group (the local σ-neighbourhood).