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where diameter(S) is the diameter of the initial swarm and the maximum radius, Rmax , is (1613) Rmax = ||xm y||, m = 1, , ns with ||xm y|| ||xi y||, i = 1, , ns (1614) is close to zero, the swarm has little potential for improvement, unless the global best is still moving In the equations above, || || is a suitable distance norm, eg Euclidean distance The algorithm is terminated when Rnorm < If is too large, the search process may stop prematurely before a good solution is found Alternatively, if is too small, the search may take excessively more iterations for the particles to form a compact swarm, tightly centered around the global best position Algorithm 163 Particle Clustering Algorithm Initialize cluster C = { }; y for about 5 times do Calculate the centroid of cluster C: x= for xi S do if ||xi x|| < then C C {xi }; end endFor endFor A more aggressive version of the radius method above can be used, where particles are clustered in the search space Algorithm 163 provides a simple particle clustering algorithm from [863] The result of this algorithm is a single cluster, C If |C|/ns > , the swarm is considered to have converged If, for example, = 07, the search will terminate if at least 70% of the particles are centered around the global best position The threshold in Algorithm 163 has the same importance as for the radius method above Similarly, if is too small, the search may terminate prematurely This clustering approach is similar to the radius approach except that the clustering approach will more readily decide that the swarm has converged Terminate when the objective function slope is approximately zero The stopping conditions above consider the relative positions of particles in the search space, and do not take into consideration information about the slope of.

(1615)

5/6/1998

16 Particle Swarm Optimization the objective function To base termination on the rate of change in the objective function, consider the ratio [863], f (t) = f ( (t)) f ( (t 1)) y y f ( (t)) y (1616)

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If f (t) < for a number of consecutive iterations, the swarm is assumed to have converged This approximation to the slope of the objective function is superior to the methods above, since it actually determines if the swarm is still making progress using information about the search space The objective function slope approach has, however, the problem that the search will be terminated if some of the particles are attracted to a local minimum, irrespective of whether other particles may still be busy exploring other parts of the search space It may be the case that these exploring particles could have found a better solution had the search not terminated To solve this problem, the objective function slope method can be used in conjunction with the radius or cluster methods to test if all particles have converged to the same point before terminating the search process In the above, convergence does not imply that the swarm has settled on an optimum (local or global) With the term convergence is meant that the swarm has reached an equilibrium, ie just that the particles converged to a point, which is not necessarily an optimum [863]

5/22/1995

The feature that drives PSO is social interaction Particles within the swarm learn from each other and, on the basis of the knowledge obtained, move to become more similar to their better neighbors The social structure for PSO is determined by the formation of overlapping neighborhoods, where particles within a neighborhood in uence one another This is in analogy with observations of animal behavior, where an organism is most likely to be in uenced by others in its neighborhood, and where organisms that are more successful will have a greater in uence on members of the neighborhood than the less successful Within the PSO, particles in the same neighborhood communicate with one another by exchanging information about the success of each particle in that neighborhood All particles then move towards some quanti cation of what is believed to be a better position The performance of the PSO depends strongly on the structure of the social network The ow of information through a social network, depends on (1) the degree of connectivity among nodes (members) of the network, (2) the amount of clustering (clustering occurs when a node s neighbors are also neighbors to one another), and (3) the average shortest distance from one node to another [892] With a highly connected social network, most of the individuals can communicate with one another, with the consequence that information about the perceived best member quickly lters through the social network In terms of optimization, this means faster

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5/29/2001

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