Advertisement

Soft Computing

, Volume 22, Issue 12, pp 4047–4070 | Cite as

Convergence analysis of standard particle swarm optimization algorithm and its improvement

  • Weiyi Qian
  • Ming Li
Methodologies and Application
  • 216 Downloads

Abstract

Standard particle swarm optimization (PSO) algorithm is a kind of stochastic optimization algorithm. Its convergence, based on probability theory, is analyzed in detail. We prove that the standard PSO algorithm is convergence with probability 1 under certain condition. Then, a new improved particle swarm optimization (IPSO) algorithm is proposed to ensure that IPSO algorithm is convergence with probability 1. In order to balance the exploration and exploitation abilities of IPSO algorithm, we propose the exploration and exploitation operators and weight the two operators in IPSO algorithm. Finally, IPSO algorithm is tested on 13 benchmark test functions and compared with the other algorithms published in the recent literature. The numerical results confirm that IPSO algorithm has the better performance in solving nonlinear functions.

Keywords

Standard particle swarm optimization Analysis of algorithm Convergence in probability Global optimization 

Notes

Acknowledgements

This work is partly supported by the National Natural Science Foundation of China (11371071) and Scientific Research Foundation of Liaoning Province Educational Department (L2013426).

Compliance with ethical standards

Conflict of interest

The authors declare that there is no conflict of interests regarding the publication of this paper.

Ethical approval

This article does not contain any studies with human participants or animals performed by any of the authors.

References

  1. Adewumi AO, Arasomwan AM (2016) An improved particle swarm optimizer based on swarm success rate for global optimization problems. J Exp Theor Artif Intell 28(3):441–483CrossRefGoogle Scholar
  2. Afshar MH (2012) Large scale reservoir operation by constrained particle swarm optimization algorithms. J Hydro-Environ Res 6(1):75–87MathSciNetCrossRefGoogle Scholar
  3. Ali MM, Kaelo P (2008) Improved particle swarm algorithms for global optimization. Appl Math Comput 196(2):578–593MathSciNetzbMATHGoogle Scholar
  4. Arasomwan AM, Adewumi AO (2014) An investigation into the performance of particle swarm optimization with various chaotic maps. Math Probl Eng pp 1–17, Article ID 178959. doi: 10.1155/2014/178959
  5. Behnamian J, Fatemi Ghomi SMT (2010) Development of a PSO-SA hybrid metaheuristic for a new comprehensive regression model to time-series forecasting. Expert Syst Appl 37(2):974–984CrossRefGoogle Scholar
  6. Chen CH, Chen YP (2011) Convergence time analysis of particle swarm optimization based on particle interaction. Adv Artif Intell pp. 1–7, Article ID 204750. doi: 10.1155/2011/204750
  7. Chuang LY, Yang CH, Li JC (2011) Chaotic maps based on binary particle swarm optimization for feature selection. Appl Soft Comput 11(1):239–248CrossRefGoogle Scholar
  8. Clerc M, Kennedy J (2002) The particle swarm: explosion, stability, and convergence in a multidimensional complex space. IEEE Trans Evol Comput 6(1):58–73CrossRefGoogle Scholar
  9. Ding J, Liu J, Chowdhury KR, Zhang W, Hu Q, Lei J (2014) A particle swarm optimization using local stochastic search and enhancing diversity for continuous optimization. Neurocomputing 137:261–267CrossRefGoogle Scholar
  10. Emara HM, Fattah AHA (2004) Continuous swarm optimization technique with stability analysis. In: Proceedings of American control conference, Boston, MA, pp 2811–2817Google Scholar
  11. Feng Y, Teng GF, Wang AX, Yao YM (2007) Chaotic inertia weight in particle swarm optimization. In: Proceedings of the 2nd international conference on innovative computing, information and control, KumamotoGoogle Scholar
  12. Fernandez-Martinez JL, Carcia-Gonzalo E, Saraswathi S, Jernigan R, Kloczkowski A (2011) Particle swarm optimization: a powerful family of stochastic optimizers. Analysis, design and application to inverse modelling. In: Proceedings of the 2nd international conference on advances in swarm intelligence. Springer, Berlin, pp 1–8Google Scholar
  13. Franken N, Engelbrecht A (2005) Particle swarm optimization approaches to coevolve strategies for the iterated prisoner’s dilemma. IIEEE Trans Evol Comput 9(6):562–579CrossRefGoogle Scholar
  14. Gandomi AH, Yun GJ, Yang XS, Talatahari S (2013) Chaos-enhanced accelerated particle swarm optimization. Commun Nonlinear Sci Numer Simul 18(2):327–340MathSciNetCrossRefzbMATHGoogle Scholar
  15. Jiang M, Luo YP, Yang SY (2007) Stochastic convergence analysis and parameter selection of the standard particle swarm optimization algorithm. Inf Process Lett 102(1):8–16MathSciNetCrossRefzbMATHGoogle Scholar
  16. Jin XL, Ma LH, Wu TJ, Qian JX (2007) Convergence analysis of the particle swarm optimization based on stochastic processes. Acta Autom Sin 33(12):1263–1268MathSciNetzbMATHGoogle Scholar
  17. Kadirkamanathan V, Selvarajajah K, Fleming PJ (2006) Stability analysis of the particle dynamics in particle swarm optimizer. IEEE Trans Evol Comput 10(3):45–255CrossRefGoogle Scholar
  18. Kennedy J, Eberhart RC (1995) Particle swarm optimization. In: IEEE international conference on neural networks, Piscataway, NJ, pp 1942–1948Google Scholar
  19. Li N, Sun DB, Zou T, Qin YQ, Wei Y (2006) An analysis for a particle’s trajectory of PSO based on difference equation. Chin J Comput 29(11):2052–2061Google Scholar
  20. Lim WH, Isa NAM (2014) Teaching and peer-learning particle swarm optimization. Appl Soft Comput 18:39–58CrossRefGoogle Scholar
  21. Malik RF, Rahman TA, Hashim SZM, Ngah R (2007) New particle swarm optimizer with sigmoid increasing inertia weight. Int J Comput Sci Secur 1(2):35–44Google Scholar
  22. Nickabadi A, Ebadzadeh MM, Safabakhsh R (2011) A novel particle swarm optimization algorithm with adaptive inertia weight. Appl Soft Comput 11:3658–3670CrossRefGoogle Scholar
  23. Ozcan, E, Mohan CK (1998) Analysis of a simple particle swarm optimization system. In: Intelligent engineering systems through artificial neural networks, pp 253–258Google Scholar
  24. Ozcan, E, Mohan CK (1999) Particle swarm optimization: surfing the waves. In: Congress on evolutionary computation. IEEE Press, Washington, DC, pp 1939–1944Google Scholar
  25. Pan F, Li XT, Zhou Q, Li WX, Gao Q (2013) Analysis of standord particle swarm optimization algorithm based on Markov chain. Acta Autom Sin 39(4):281–289Google Scholar
  26. Rapaic MR, Kanovic Z (2009) Time-varying PSO-convergence analysis, convergence-related parameterization and new parameter adjustment schemes. Inf Process Lett 109(11):548–552MathSciNetCrossRefzbMATHGoogle Scholar
  27. Shen X, Chi Z, Yang J, Chen C, Chi Z (2010) Particle swarm optimization with dynamic adaptive inertia weight. In: International conference on challenges in environmental science and computer engineering, Wuhan, pp 287–290Google Scholar
  28. Shi YH, Eberhart RC (1999) Empirical study of particle swarm optimization. In: IEEE international conference on evolutionary computation, Washington, DC, pp 1945–1950Google Scholar
  29. Trelea IC (2003) The particle swarm optimization algorithm: convergence analysis and parameter selection. Inf Process Lett 85(6):317–325MathSciNetCrossRefzbMATHGoogle Scholar
  30. van den Bergh F (2002) An analysis of particle swarm optimization. PhD thesis, Department of Computer Science, University of PretoriaGoogle Scholar
  31. van den Bergh F, Engelbrecht AP (2006) A study of particle swarm optimization particle trajectories. Inf Sci 176(8):937–971MathSciNetCrossRefzbMATHGoogle Scholar
  32. Victoire A, Jeyakumar AE (2005) Reserve constrained dynamic dispatch of units with valve-point effects. IEEE Trans Power Syst 20(3):1273–1282CrossRefGoogle Scholar
  33. Wang L, Yang B, Chen Y (2014) Improving particle swarm optimization using multi-layer searching strategy. Inf Sci 274:70–94CrossRefGoogle Scholar
  34. Yao X, Liu Y, Lin G (1999) Evolutionary programming made faster. IEEE Trans Evol Comput 3(2):81–102Google Scholar
  35. Yu X, Zhang X (2014) Enhanced comprehensive learning particle swarm optimization. Appl Math Comput 242:265–276MathSciNetzbMATHGoogle Scholar
  36. Zou F, Wang L, Hei X, Chen D, Yang D (2014) Teaching-learning-based optimization with dynamic group strategy for global optimization. Inf Sci 273:112–131CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.College of Mathematics and PhysicsBohai UniversityJinzhouChina

Personalised recommendations