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Particle Swarm Optimization: Understanding Order-2 Stability Guarantees

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Applications of Evolutionary Computation (EvoApplications 2019)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11454))

Abstract

This paper’s primary aim is to provide clarity on which guarantees about particle stability can actually be made. The particle swarm optimization algorithm has undergone a considerable amount of theoretical analysis. However, with this abundance of theory has come some terminological inconstancies, and as a result it is easy for a practitioner to be misguided by overloaded terminology. Specifically, the criteria for both order-1 and order-2 stability are well studied, but the exact definition of order-2 stability is not consistent amongst researchers. A consequence of this inconsistency in terminology is that the existing theory may in fact misguide practitioners instead of assisting them. In this paper it is theoretically and empirically demonstrated which practical guarantees can in fact be made about particle stability. Specifically, it is shown that the definition of order-2 stability which accurately reflects PSO behavior is that of convergence in second order moment to a constant, and not to zero.

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References

  1. Kennedy, J., Eberhart, R.: Particle swarm optimization. In: Proceedings of the IEEE International Joint Conference on Neural Networks, pp. 1942–1948. IEEE Press, Piscataway, NJ (1995)

    Google Scholar 

  2. Poli, R.: Analysis of the publications on the applications of particle swarm optimisation. J. Artif. Evol. Appl. 2008, 1–10 (2008)

    Google Scholar 

  3. Ozcan, E., Mohan, C.K.: Analysis of a simple particle swarm optimization system. Intell. Eng. Syst. Artif. Neural Netw. 8, 253–258 (1998)

    Google Scholar 

  4. Clerc, M., Kennedy, J.: The particle swarm-explosion, stability, and convergence in a multidimensional complex space. IEEE Trans. Evol. Comput. 6(1), 58–73 (2002)

    Article  Google Scholar 

  5. Jiang, M., Luo, Y.P., Yang, S.Y.: Stochastic convergence analysis and parameter selection of the standard particle swarm optimization algorithm. Inf. Process. Lett. 102(1), 8–16 (2007)

    Article  MathSciNet  Google Scholar 

  6. García-Gonzalo, E., Fernández-Martinez, J.L.: Convergence and stochastic stability analysis of particle swarm optimization variants with generic parameter distributions. Appl. Math. Comput. 249, 286–302 (2014)

    MathSciNet  MATH  Google Scholar 

  7. Cleghorn, C.W., Engelbrecht, A.P.: A generalized theoretical deterministic particle swarm model. Swarm Intell. 8(1), 35–59 (2014)

    Article  Google Scholar 

  8. Liu, Q.: Order-2 stability analysis of particle swarm optimization. Evol. Comput. 1–30 (2014). https://doi.org/10.1162/EVCO_a_00129

  9. Cleghorn, C.W., Engelbrecht, A.P.: Particle swarm stability: a theoretical extension using the non-stagnate distribution assumption. Swarm Intell. 12(1), 1–22 (2018)

    Article  Google Scholar 

  10. Trelea, I.C.: The particle swarm optimization algorithm: convergence analysis and parameter selection. Inf. Process. Lett. 85(6), 317–325 (2003)

    Article  MathSciNet  Google Scholar 

  11. Van den Bergh, F., Engelbrecht, A.P.: A study of particle swarm optimization particle trajectories. Inf. Sci. 176(8), 937–971 (2006)

    Article  MathSciNet  Google Scholar 

  12. Cleghorn, C.W., Engelbrecht, A.P.: Particle swarm optimizer: the impact of unstable particles on performance. In: Proceedings of the IEEE Symposium Series on Swarm Intelligence, pp. 1–7. IEEE Press, Piscataway, NJ (2016)

    Google Scholar 

  13. Cleghorn, C.W., Engelbrecht, A.P.: Fully informed particle swarm optimizer: convergence analysis. In: Proceedings of the IEEE Congress on Evolutionary Computation, pp. 164–170. IEEE Press, Piscataway, NJ (2015)

    Google Scholar 

  14. Cleghorn, C.W., Engelbrecht, A.P.: Unified particle swarm optimizer: convergence analysis. In: Proceedings of the IEEE Congress on Evolutionary Computation, pp. 448–454. IEEE Press, Piscataway, NJ (2016)

    Google Scholar 

  15. Cleghorn, C.W., Engelbrecht, A.P.: Fitness-distance-ratio particle swarm optimization: stability analysis. In: Proceedings of the Genetic and Evolutionary Computation Conference, pp. 12–18. ACM Press, New York, NY (2017)

    Google Scholar 

  16. Cleghorn, C.W., Engelbrecht, A.P.: Particle swarm variants: standardized convergence analysis. Swarm Intell. J. 9(2–3), 177–203 (2015)

    Article  Google Scholar 

  17. Bonyadi, M.R., Michalewicz, Z.: Analysis of stability, local convergence, and transformation sensitivity of a variant of particle swarm optimization algorithm. IEEE Trans. Evol. Comput. 20(3), 370–385 (2016)

    Article  Google Scholar 

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

    Google Scholar 

  19. Engelbrecht, A.P.: Particle swarm optimization: Global best or local best. In: Proceedings of the 1st BRICS Countries Congress on Computational Intelligence, pp. 124–135. IEEE Press, Piscataway, NJ (2013)

    Google Scholar 

  20. Bonyadi, M.R., Michalewicz, Z.: Particle swarm optimization for single objective continuous space problems: a review. Evol. Comput. 25(1), 1–54 (2016)

    Article  Google Scholar 

  21. Ozcan, E., Mohan, C.K.: Particle swarm optimization: surfing the waves. In: Proceedings of the IEEE Congress on Evolutionary Computation, vol. 3, pp. 1939–1944. IEEE Press, Piscataway, NJ (July 1999)

    Google Scholar 

  22. Kadirkamanathan, V., Selvarajah, K., Fleming, P.J.: Stability analysis of the particle dynamics in particle swarm optimizer. IEEE Trans. Evol. Comput. 10(3), 245–255 (2006)

    Article  Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

  25. Blackwell, T.: A study of collapse in bare bones particle swarm optimization. IEEE Trans. Evol. Comput. 16(3), 354–372 (2012)

    Article  Google Scholar 

  26. Bonyadi, M.R., Michalewicz, Z.: Stability analysis of the particle swarm optimization without stagnation assumption. IEEE Trans. Evol. Comput. PP(1), 1–7 (2015)

    Google Scholar 

  27. Poli, R.: Mean and variance of the sampling distribution of particle swarm optimizers during stagnation. IEEE Trans. Evol. Comput. 13(4), 712–721 (2009)

    Article  Google Scholar 

  28. Engelbrecht, A.P.: Roaming behavior of unconstrained particles. In: Proceedings of the 1st BRICS Countries Congress on Computational Intelligence, pp. 104–111. IEEE Press, Piscataway, NJ (2013)

    Google Scholar 

  29. Liang, J.J.K., Qu, B.Y., Suganthan, P.N.: Problem definitions and evaluation criteria for the CEC 2014 special session and competition on single objective real-parameter numerical optimization. Technical report 201311, Computational Intelligence Laboratory, Zhengzhou University and Nanyang Technological University (2013)

    Google Scholar 

  30. Kreyszig, E.: Introductory Functional Analysis with Applications. Wiley, Wiltshire (1978)

    MATH  Google Scholar 

  31. Cleghorn, C.W., Engelbrecht, A.P.: Particle swarm convergence: an empirical investigation. In: Proceedings of the Congress on Evolutionary Computation, pp. 2524–2530. IEEE Press, Piscataway, NJ (2014)

    Google Scholar 

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Author information

Authors and Affiliations

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

    Christopher W. Cleghorn

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  1. Christopher W. Cleghorn
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Correspondence to Christopher W. Cleghorn .

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Editors and Affiliations

  1. University of Mainz, Mainz, Germany

    Paul Kaufmann

  2. University of Granada, Granada, Spain

    Pedro A. Castillo

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Cleghorn, C.W. (2019). Particle Swarm Optimization: Understanding Order-2 Stability Guarantees. In: Kaufmann, P., Castillo, P. (eds) Applications of Evolutionary Computation. EvoApplications 2019. Lecture Notes in Computer Science(), vol 11454. Springer, Cham. https://doi.org/10.1007/978-3-030-16692-2_36

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  • DOI: https://doi.org/10.1007/978-3-030-16692-2_36

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-16691-5

  • Online ISBN: 978-3-030-16692-2

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