The Particle Swarm—Explosion, Stability, and Convergence in a Multidimensional Complex Space

发布时间：2012-03-18 11:25:44

58IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION,VOL.6,NO.1,FEBRUARY2002 The Particle Swarm—Explosion,Stability,and Convergence in a Multidimensional Complex SpaceMaurice Clerc and James KennedyAbstract—The particle swarm is an algorithm for finding op-timal regions of complex search spaces through the interaction of individuals in a population of particles.Even though the algorithm, which is based on a metaphor of social interaction,has been shown to perform well,researchers have not adequately explained how it works.Further,traditional versions of the algorithm have had some undesirable dynamical properties,notably the particles’ve-locities needed to be limited in order to control their trajectories. The present paper analyzes a particle’s trajectory as it moves in discrete time(the algebraic view),then progresses to the view of it in continuous time(the analytical view).A five-dimensional de-piction is developed,which describes the system completely.These analyses lead to a generalized model of the algorithm,containing a set of coefficients to control the system’s convergence tendencies. Some results of the particle swarm optimizer,implementing modi-fications derived from the analysis,suggest methods for altering the original algorithm in ways that eliminate problems and increase the ability of the particle swarm to find optima of some well-studied test functions.Index Terms—Convergence,evolutionary computation,opti-mization,particle swarm,stability.I.I NTRODUCTIONP ARTICLE swarm adaptation has been shown to suc-cessfully optimize a wide range of continuous functions [1]–[5].The algorithm,which is based on a metaphor of social interaction,searches a space by adjusting the trajectories of individual vectors,called“particles”as they are conceptualized as moving points in multidimensional space.The individual particles are drawn stochastically toward the positions of their own previous best performance and the best previous performance of their neighbors.While empirical evidence has accumulated that the algorithm “works,”e.g.,it is a useful tool for optimization,there has thus far been little insight into how it works.The present analysis begins with a highly simplified deterministic version of the par-ticle swarm in order to provide an understanding about how it searches the problem space[4],then continues on to analyze the full stochastic system.A generalized model is proposed,in-cluding methods for controlling the convergence properties of the particle system.Finally,some empirical results are given, showing the performance of various implementations of the al-gorithm on a suite of test functions.Manuscript received January24,2000;revised October30,2000and April 30,2001.M.Clerc is with the France Télécom,74988Annecy,France(e-mail:Maurice. Clerc@WriteMe.com).J.Kennedy is with the Bureau of Labor Statistics,Washington,DC20212 USA(e-mail:Kennedy_jim@bls.gov).Publisher Item Identifier S1089-778X(02)02209-9.A.The Particle SwarmA population of particles is initialized with randompositionsand afunction

The Particle Swarm—Explosion, Stability, and Convergence in a Multidimensional Complex Space

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