Particle swarm optimisation (PSO) is a widely used meta-heuristic algorithm, which has been successfully applied to a large variety of problems in optimisation, prediction, classification, visualisation and robotics. However, the underlying mechanism that leads to good solutions in these domains is not well understood. A number of parameters influence the algorithm mainly by determining the balance between exploration and exploitation. In this chapter, we describe the behaviour of PSO as a product of random matrices, and use a Lyapunov exponent to precisely characterise the critical parameters for PSO. The theoretical results are discussed based on numerical experiments for standard benchmark problems.

Chapter Contents:

• Abstract
• 10.1 Introduction
• 10.1.1 Particle swarm optimisation
• 10.1.1.1 PSO parameter selection and resultant behaviours
• 10.1.1.2 Alternative meta-heuristics
• 10.1.1.3 The problem of making comparisons between algorithms
• 10.1.2 On criticality
• 10.1.3 The dynamics of PSO
• 10.1.4 Matrix formulation
• 10.2 Critical swarm conditions for a single particle
• 10.2.1 PSO as a random dynamical system
• 10.2.2 Stability
• 10.2.3 Personal best vs. global best
• 10.3 Simulations
• 10.3.1 Combining the results from multiple cost functions
• 10.4 Discussion
• 10.4.1 Relevance of criticality
• 10.4.2 Comparison with earlier explanations
• 10.4.2.1 Lines of stability when viewed against average cost function results
• 10.4.2.2 Lines of stability when viewed against individual cost function results
• 10.4.2.3 Lines of stability when viewed against detailed cost function results
• 10.4.3 Switching dynamics at discovery of better solutions
• 10.4.4 The role of personal best, present best and best ever
• 10.5 Conclusion
• Acknowledgement
• References

Inspec keywords: particle swarm optimisation

Other keywords: PSO; meta-heuristic algorithm; particle swarm optimisation; critical parameters

Subjects: Optimisation; Optimisation techniques; Optimisation techniques