Theoretical Analysis of Initial Particle Swarm Behavior

  • Sabine Helwig
  • Rolf Wanka
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5199)


In this paper, particle trajectories of PSO algorithms in the first iteration are studied. We will prove that many particles leave the search space at the beginning of the optimization process when solving problems with boundary constraints in high-dimensional search spaces. Three different velocity initialization strategies will be investigated, but even initializing velocities to zero cannot prevent this particle swarm explosion. The theoretical analysis gives valuable insight into PSO in high-dimensional bounded spaces, and highlights the importance of bound handling for PSO: As many particles leave the search space in the beginning, bound handling strongly influences particle swarm behavior. Experimental investigations confirm the theoretical results.


Particle Swarm Optimization Search Space Particle Swarm Particle Swarm Optimization Algorithm Standard Particle Swarm Optimization 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Kennedy, J., Eberhart, R.C.: Swarm Intelligence. Morgan Kaufmann, San Francisco (2001)Google Scholar
  2. 2.
    Clerc, M.: Confinements and Biases in Particle Swarm Optimization (2006),
  3. 3.
    Alvarez-Benitez, J.E., Everson, R.M., Fieldsend, J.E.: A MOPSO Algorithm Based Exclusively on Pareto Dominance Concepts. In: Evolutionlary Multi-Criterion Optimization, pp. 459–473 (2005)Google Scholar
  4. 4.
    Zhang, W.J., Xie, X.F., Bi, D.C.: Handling Boundary Constraints for Numerical Optimization by Particle Swarm Flying in Periodic Search Space. In: Proceedings of the IEEE Congress on Evol. Computation, vol. 2, pp. 2307–2311 (2004)Google Scholar
  5. 5.
    Robinson, J., Rahmat-Samii, Y.: Particle Swarm Optimization in Electromagnetics. IEEE Transactions on Antennas and Propagation 52(2), 397–407 (2004)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Bratton, D., Kennedy, J.: Defining a Standard for Particle Swarm Optimization. In: Proc. IEEE Swarm Intelligence Symp., pp. 120–127 (2007)Google Scholar
  7. 7.
    Clerc, M.: Particle Swarm Optimization. ISTE Ltd (2006)Google Scholar
  8. 8.
    Clerc, M., Kennedy, J.: The Particle Swarm - Explosion, Stability, and Convergence in a Multidimensional Complex Space. IEEE Transactions on Evolutionary Computation 6(1), 58–73 (2002)CrossRefGoogle Scholar
  9. 9.
    Trelea, I.C.: The particle swarm optimization algorithm: convergence analysis and parameter selection. Information Processing Letters 85(6), 317–325 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Jiang, M., Luo, Y.P., Yang, S.Y.: Stochastic convergence analysis and parameter selection of the standard particle swarm optimization algorithm. Information Processing Letters 102(1), 8–16 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Clerc, M., et al.: Standard PSO 2007 (2007), (standard_pso_2007.c)
  12. 12.
    Helwig, S., Wanka, R.: Particle Swarm Optimization in High-Dimensional Bounded Search Spaces. In: Proc. IEEE Swarm Intelligence Symp., pp. 198–205 (2007)Google Scholar
  13. 13.
    Engelbrecht, A.P.: Fundamentals of Computational Swarm Intelligence. John Wiley and Sons Ltd, Chichester (2005)Google Scholar
  14. 14.
    Bradley, D., Gupta, R.: On the Distribution of the Sum of n Non-Identically Distributed Uniform Random Variables. Annals of the Institute of Statistical Mathematics 54(3), 689–700 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Suganthan, P.N., Hansen, N., Liang, J.J., Deb, K., Chen, Y., Auger, A., Tiwari, S.: Problem Definitions and Evaluation Criteria for the CEC 2005 Special Session on Real-Parameter Optimization. KanGAL Report 2005005, Nanyang Technological University, Singapore (2005)Google Scholar
  16. 16.
    Kennedy, J., Mendes, R.: Population Structure and Particle Swarm Performance. In: Proceedings of the IEEE Congress on Evol. Computation, pp. 1671–1676 (2002)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Sabine Helwig
    • 1
  • Rolf Wanka
    • 1
  1. 1.Department of Computer ScienceUniversity of Erlangen-NurembergGermany

Personalised recommendations