Swarm Intelligence

, Volume 9, Issue 4, pp 291–314 | Cite as

Critical considerations on angle modulated particle swarm optimisers

  • Barend J. Leonard
  • Andries P. Engelbrecht
  • Christopher W. Cleghorn


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 has 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.


Swarm intelligence Particle swarm optimisation Angle modulation Discrete optimisation Binary optimisation 



The authors would like to thank the reviewers of this article for their valuable comments and suggestions.


  1. Engelbrecht, A. (2012). Particle swarm optimization: Velocity initialization. In IEEE congress on evolutionary computation (pp. 1–8).Google Scholar
  2. Franken, N. (2004). PSO-based coevolutionary game learning. Master’s thesis, University of PretoriaGoogle Scholar
  3. Holland, J. (1975). Adaptation in natural and artificial systems. Ann Arbor, MI: The University of Michigan Press.Google Scholar
  4. Kennedy, J., & Eberhart, R. (1995). Particle swarm optimization. In Proceedings of the IEEE international conference on neural networks (Vol. 4, pp. 1942–1948)Google Scholar
  5. Kennedy, J., & Eberhart, R. (1997). A discrete binary version of the particle swarm algorithm. In IEEE international conference on systems, man, and cybernetics. computational cybernetics and simulation (Vol. 5, pp. 4104–4108)Google Scholar
  6. Kirkpatrick, S., Gelatt, C., & Vecchi, M. (1983). Optimization by simulated annealing. Science, 220(4598), 671–680.zbMATHMathSciNetCrossRefGoogle Scholar
  7. Leonard, B., & Engelbrecht, A. (2014). Angle modulated particle swarm variants. In Swarm intelligence, LNCS (Vol. 8667 pp. 38–49).Google Scholar
  8. Pampara, G. (2013) Angle modulated population based algorithms to solve binary problems. Master’s thesis, University of PretoriaGoogle Scholar
  9. Pampara, G., Franken, N., & Engelbrecht, A. (2005). Combining particle swarm optimisation with angle modulation to solve binary problems. In IEEE congress on evolutionary computation, 2005 (Vol. 1, pp. 89–96)Google Scholar
  10. Pampara, G., Engelbrecht, A., & Franken, N. (2006). Binary differential evolution. In IEEE congress on evolutionary computation (pp. 1873–1879)Google Scholar
  11. Poli, R. (2009). Mean and variance of the sampling distribution of particle swarm optimizers during stagnation. IEEE Transactions on Evolutionary Computation, 13(4), 712–721.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Barend J. Leonard
    • 1
  • Andries P. Engelbrecht
    • 1
  • Christopher W. Cleghorn
    • 1
  1. 1.Department of Computer ScienceUniversity of PretoriaPretoriaSouth Africa

Personalised recommendations