Particles with Spatial Extention

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Particles with spatial extension were developed to prevent the swarm from prematurely converging [489, 877] If one particle locates an optimum, then all particles will be attracted to the optimum causing all particles to cluster closely The spatial extension of particles allows some particles to explore other areas of the search space, while others converge to the optimum to further re ne it The exploring particles may locate a di erent, more optimal solution The objective of spatial extension is to dynamically increase diversity when particles start to cluster This is achieved by adding a radius to each particle If two particles collide, ie their radii intersect, then the two particles bounce o Krink et al [489] and Vesterstr m and Riget [877] investigated three strategies for spatial extension: random bouncing, where colliding particles move in a random new direction at the same speed as before the collision;

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340 realistic physical bouncing; and

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16 Particle Swarm Optimization

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simple velocity-line bouncing, where particles continue to move in the same direction but at a scaled speed With scale factor in [0, 1] particles slow down, while a scale factor greater than one causes acceleration to avoid a collision A negative scale factor causes particles to move in the opposite direction to their previous movement Krink et al showed that random bouncing is not as e cient as the consistent bouncing methods To ensure convergence of the swarm, particles should bounce o on the basis of a probability An initial large bouncing probability will ensure that most collisions result in further exploration, while a small bouncing probability for the nal steps will allow the swarm to converge At all times, some particles will be allowed to cluster together to re ne the current best solution

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Binary PSO

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PSO was originally developed for continuous-valued search spaces Kennedy and Eberhart developed the rst discrete PSO to operate on binary search spaces [450, 451] Since real-valued domains can easily be transformed into binary-valued domains (using standard binary coding or Gray coding), this binary PSO can also be applied to realvalued optimization problems after such transformation (see [450, 451] for applications of the binary PSO to real-valued problems) For the binary PSO, particles represent positions in binary space Each element of a particle s position vector can take on the binary value 0 or 1 Formally, xi Bnx , or xij {0, 1} Changes in a particle s position then basically implies a mutation of bits, by ipping a bit from one value to the other A particle may then be seen to move to near and far corners of a hypercube by ipping bits One of the rst problems to address in the development of the binary PSO, is how to interpret the velocity of a binary vector Simply seen, velocity may be described by the number of bits that change per iteration, which is the Hamming distance between xi (t) and xi (t + 1), denoted by H(xi (t), xi (t + 1)) If H(xi (t), xi (t + 1)) = 0, zero bits are ipped and the particle does not move; ||vi (t)|| = 0 On the other hand, ||vi (t)|| = nx is the maximum velocity, meaning that all bits are ipped That is, xi (t + 1) is the complement of xi (t) Now that a simple interpretation of the velocity of a bit-vector is possible, how is the velocity of a single bit (single dimension of the particle) interpreted In the binary PSO, velocities and particle trajectories are rather de ned in terms of probabilities that a bit will be in one state or the other Based on this probabilistic view, a velocity vij (t) = 03 implies a 30% chance to be bit 1, and a 70% chance to be bit 0 This means that velocities are restricted to be in the range [0, 1] to be interpreted as a probability Di erent methods can be employed to normalize velocities such that vij [0, 1] One approach is to simply divide each vij by the maximum velocity, Vmax,j While this approach will ensure velocities are in the range [0,1], consider

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