Soft Computing

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

Convergence analysis of standard particle swarm optimization algorithm and its improvement

  • Weiyi QianEmail author
  • Ming Li
Methodologies and Application


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.


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



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.


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© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.College of Mathematics and PhysicsBohai UniversityJinzhouChina

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