Convergence analysis of standard particle swarm optimization algorithm and its improvement
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.
KeywordsStandard 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.
This article does not contain any studies with human participants or animals performed by any of the authors.
- 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
- 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
- Emara HM, Fattah AHA (2004) Continuous swarm optimization technique with stability analysis. In: Proceedings of American control conference, Boston, MA, pp 2811–2817Google Scholar
- 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
- 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
- Kennedy J, Eberhart RC (1995) Particle swarm optimization. In: IEEE international conference on neural networks, Piscataway, NJ, pp 1942–1948Google Scholar
- 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
- 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
- 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
- Ozcan, E, Mohan CK (1999) Particle swarm optimization: surfing the waves. In: Congress on evolutionary computation. IEEE Press, Washington, DC, pp 1939–1944Google Scholar
- 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
- 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
- Shi YH, Eberhart RC (1999) Empirical study of particle swarm optimization. In: IEEE international conference on evolutionary computation, Washington, DC, pp 1945–1950Google Scholar
- van den Bergh F (2002) An analysis of particle swarm optimization. PhD thesis, Department of Computer Science, University of PretoriaGoogle Scholar
- Yao X, Liu Y, Lin G (1999) Evolutionary programming made faster. IEEE Trans Evol Comput 3(2):81–102Google Scholar