Skip to main content
SpringerLink
Log in

A generalized theoretical deterministic particle swarm model

  • Regular Paper
  • Published:
Swarm Intelligence Aims and scope Submit manuscript

Abstract

A number of theoretical studies of particle swarm optimization (PSO) have been done to gain a better understanding of the dynamics of the algorithm and the behavior of the particles under different conditions. These theoretical analyses have been performed for both the deterministic PSO model and more recently for the stochastic model. However, all current theoretical analyses of the PSO algorithm were based on the stagnation assumption, in some form or another. The analysis done under the stagnation assumption is one where the personal best and neighborhood best positions are assumed to be non-changing. While analysis under the stagnation assumption is very informative, it could never provide a complete description of a PSO’s behavior. Furthermore, the assumption implicitly removes the notion of a social network structure from the analysis. This paper presents a generalization to the theoretical deterministic PSO model. Under the generalized model, conditions for particle convergence to a point are derived. The model used in this paper greatly weakens the stagnation assumption, by instead assuming that each particle’s personal best and neighborhood best can occupy an arbitrarily large number of unique positions. It was found that the conditions derived in previous theoretical deterministic PSO research could be obtained as a specialization of the new generalized model proposed. Empirical results are presented to support the theoretical findings.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11

Similar content being viewed by others

References

  • Burden, R., & Faires, J. (2010). Numerical analysis (9th ed.). Hampshire: Cengage Learning.

    Google Scholar 

  • Carlisle, A., & Dozier, G. (2001). An off-the-shelf PSO. In Proceedings of the Workshop on Particle Swarm Optimization (pp. 1–6). Indianapolis, IN: IUPUI.

  • Clerc, M. (1999). The swarm and the queen: Towards a deterministic and adaptive particle swarm optimization. In Proceedings of the IEEE Congress on Evolutionary Computation (Vol. 3, pp. 1951–1957). Piscataway, NJ: IEEE Press.

  • Clerc, M., & Kennedy, J. (2002). The particle swarm-explosion, stability, and convergence in a multidimensional complex space. IEEE Transactions on Evolutionary Computation, 6(1), 58–73.

    Article  Google Scholar 

  • Dosoer, C., & Vidyasagar, M. (1975). Feedback systems: Input output properties. New York: Academic Press.

    Google Scholar 

  • Eberhart, R., & Shi, Y. (2000). Comparing inertia weights and constriction factors in particle swarm optimization. In Proceedings of the IEEE Congress on Evolutionary Computation (Vol. 1, pp. 84–88). Piscataway, NJ: IEEE Press.

  • Eberhart, R., Simpson, P., & Dobbins, R. (1996). Computational intelligence PC tools (1st ed.). Boston: Academic Press Professional.

    Google Scholar 

  • Engelbrecht, A. (2013a). Particle swarm optimization: Global best or local best. In 1st BRICS Countries Congress on Computational Intelligence. Piscataway, NJ: IEEE Press.

  • Engelbrecht, A. (2013b). Particle swarm optimization: Iteration strategies revisted. In 1st BRICS Countries Congress on Computational Intelligence. Piscataway, NJ: IEEE Press.

  • Gazi, V. (2012). Stochastic stability analysis of the particle dynamics in the PSO algorithm. In Proceedings of the IEEE International Symposium on Intelligent Control (pp. 708–713). Dubrovnik: IEEE Press.

  • Jiang, M., Luo, Y., & Yang, S. (2007). Stochastic convergence analysis and parameter selection of the standard particle swarm optimization algorithm. Information Processing Letters, 102, 8–16.

    Article  MATH  MathSciNet  Google Scholar 

  • Kadirkamanathan, V., Selvarajah, K., & Fleming, P. (2006). Stability analysis of the particle dynamics in particle swarm optimizer. IEEE Transactions on Evolutionary Computation, 10(3), 245–255.

    Article  Google Scholar 

  • Kennedy, J. (1999). Small worlds and mega-minds: Effects of neighborhood topology on particle swarm performance. In Proceedings of the IEEE Congress on Evolutionary Computation (Vol. 3, pp. 1931–1938). Piscataway, NJ: IEEE Press.

  • Kennedy, J., & Eberhart, R. (1995). Particle swarm optimization (pp. 1942–1948). Piscataway, NJ: IEEE Press.

    Google Scholar 

  • Kennedy, J., & Mendes, R. (2002). Population structure and particle performance. In Proceedings of the IEEE Congress on Evolutionary Computation (pp. 1671–1676). Piscataway, NJ: IEEE Press.

  • Kisacanin, B., & Agarwal, G. (2001). Linear control systems: With solved problems and Matlab examples. New York: Springer.

    Book  Google Scholar 

  • Martinez, F., Gonzalo, G., & Alvarez, F. (2008). Theoretical analysis of particle swarmtrajectories through a mechanical analogy. International Journal of Computational Intelligence Research, 4(2), 93–104.

    MathSciNet  Google Scholar 

  • Mate, L. (1999). On infinite composition of affine mappings. Fundamenta Mathematicae, 159, 85–90.

    MATH  MathSciNet  Google Scholar 

  • Naka, S., Genji, T., Yura, T., & Fukuyama, Y. (2001). Practical distribution state estimation using hybrid particle swarm optimization. In IEEE Power Engineering Society Winter Meeting (Vol. 2, pp. 815–820). Piscataway, NJ: IEEE Press.

  • Ozcan, E., & Mohan, C. (1998). Analysis of a simple particle swarm optimization system. Intelligent Engineering Systems Through Artificial Neural Networks, 8, 253–258.

    Google Scholar 

  • Ozcan, E., & Mohan, C. (1999). Particle swarm optimization: Surfing the waves. In Proceedings of the IEEE Congress on Evolutionary Computation (Vol. 3). Piscataway, NJ: IEEE Press.

  • Peer, E., Van den Bergh, F., & Engelbrecht, A. (2003). Using neighborhoods with the guaranteed convergence PSO. In Proceedings of the IEEE Swarm Intelligence Symposium (pp. 235–242). Piscataway, NJ: IEEE Press.

  • Peng, J., Chen, Y., & Eberhart, R. (2000). Battery pack state of charge estimator design using computational intelligence approaches. In Proceedings of the Annual Battery Conference on Applications and Advances (pp. 173–177). Piscataway, NJ: IEEE Press.

  • Peram, T., Veeramachaneni, K., & Mohan, C. (2003). Fitness-distance-ratio based particle swarm optimization. In Proceedings of the IEEE Swarm Intelligence Symposium (pp. 174–181). Piscataway, NJ: IEEE Press.

  • 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.

    Article  Google Scholar 

  • Poli, R., & Broomhead, D. (2007). Exact analysis of the sampling distribution for the canonical particle swarm optimiser and its convergence during stagnation. In Genetic and Evolutionary Computation Conference (pp. 134–141). New York: ACM Press.

  • Ratnaweera, A., Halgamuge, S., & Watson, H. (2003). Particle swarm optimization with self-adaptive acceleration coefficients. In Proceedings of the First International Conference on Fuzzy Systems and Knowledge Discovery (pp. 264–268). Los Vaqueros, CA: IEEE Press.

  • Shi, Y., & Eberhart, R. (1998). A modified particle swarm optimizer. In Proceedings of the IEEE Congress on Evolutionary Computation (pp. 69–73). Piscataway, NJ: IEEE Press.

  • Shi, Y., & Eberhart, R. (2001). Fuzzy adaptive particle swarm optimization. In Proceedings of the IEEE Congress on Evolutionary Computation (Vol. 1, pp. 101–106). Piscataway, NJ: IEEE Press.

  • Trelea, I. (2003). The particle swarm optimization algorithm: Convergence analysis and parameter selection. Information Processing Letters, 85(6), 317–325.

    Article  MATH  MathSciNet  Google Scholar 

  • Van den Bergh, F. (2002). An analysis of particle swarm optimizers. PhD thesis, Department of Computer Science, University of Pretoria, Pretoria, South Africa.

  • Van den Bergh, F., & Engelbrecht, A. (2006). A study of particle swarm optimization particle trajectories. Information Sciences, 176(8), 937–971.

    Article  MATH  MathSciNet  Google Scholar 

  • Venter, G., & Sobieszczanski-Sobieski, J. (2003). Particle swarm optimization. Journal for the American Institute of Aeronautics and Astronautics, 41(8), 1583–1589.

    Article  Google Scholar 

  • Vidyasagar, M. (2002). Nonlinear systems analysis. Philadelphia: SIAM.

    Book  MATH  Google Scholar 

  • Zheng, Y., Ma, L., Zhang, L., & Qian, J. (2003). On the convergence analysis and parameter selection in particle swarm optimization. In International Conference on Machine Learning and Cybernetics (Vol. 3, pp. 1802–1807). Piscataway, NJ: IEEE Press.

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, University of Pretoria, Pretoria, South Africa

    Christopher W. Cleghorn & Andries P. Engelbrecht

Authors
  1. Christopher W. Cleghorn
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Andries P. Engelbrecht
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Andries P. Engelbrecht.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Cleghorn, C.W., Engelbrecht, A.P. A generalized theoretical deterministic particle swarm model. Swarm Intell 8, 35–59 (2014). https://doi.org/10.1007/s11721-013-0090-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11721-013-0090-y

Keywords

Search

Navigation