The particle swarm optimization algorithm

The particle swarm optimization algorithm: convergence analysis and parameter selection Ioan Cristian Trelea INA P-G, UMR Génie et Microbiologie des Procédés Alimentaires, BP 01, 78850 Thiverval-Grignon, France Received 10 July 2002; received in revised form 12 September 2002 Communicated by S. Albers Abstract The particle swarm optimization algorithm is analyzed using standard results from the dynamic system theory. Graphical parameter selection guidelines are derived. The exploration–exploitation tradeoff is discussed and illustrated. Examples of

performance on benchmark functions superior to previously published results are given. 2002 Elsevier Science B.V. All rights reserved. Keywords: Particle swarm optimization; Stochastic optimization; Analysis of algorithms; Parallel algorithms 1. Introduction The particle swarm optimization (PSO) is a paral- lel evolutionary computation technique developed by Kennedy and Eberhart [4] based on the social behav- ior metaphor. A standard textbook on PSO, treating both the social and computational paradigms, is [5]. The PSO algorithm is initialized with a population of random candidate solutions, conceptualized as parti- cles. Each particle is assigned a randomized velocity and is iteratively moved through the problem space. It is attracted towards the location of the best fitness achieved so far by the particle itself and by the loca- tion of the best fitness achieved so far across the whole population (global version of the algorithm). The PSO algorithm includes some tuning parame- ters that greatly influence the algorithm performance, E-mail address: trelea@grignon.inra.fr (I.C. Trelea). often stated as the exploration–exploitation tradeoff: Exploration is the ability to test various regions in the problem space in order to locate a good optimum, hopefully the global one. Exploitation is the ability to concentrate the search around a promising candidate solution in order to locate the optimum precisely. De- spite recent research efforts, the selection of the al- gorithm parameters remains empirical to a large ex- tent. A complete theoretical analysis of the algorithm has been given by Clerc and Kennedy [2]. Based on this analysis, the authors derived a reasonable set of tuning parameters, as confirmed by [3]. The reference [2] contains a good deal of mathematical complex- ity, however, and deriving from it simple user-oriented guidelines for the parameter selection in a specific problem is not straightforward. The present work gives some additional insight into the PSO parameter selection topic. It is estab- lished that some of the parameters add no flexibil- ity to the algorithm and can be discarded without 0020-0190/02/$ – see front matter  2002 Elsevier Science B.V. All rights reserved. PII: S0020-0190(02)00447-7 318 I.C. Trelea / Information Processing Letters 85 (2003) 317–325 loss of generality. Results from the dynamic system theory are used for a relatively simple theoretical analysis of the algorithm which results in graphical guidelines for parameter selection. The user can thus take well-informed decisions according to the desired exploration–exploitation tradeoff: either favor explo- ration by a thorough sampling of the solution...

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