Cooperative PSO in Java

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The PSO algorithms discussed thus far solve optimization problems by using J-dimensional particles, where J is the number of parameters to be optimized, i.e. the number of components of the final solution. A swarm therefore has the task of finding optimal values for all J parameters. A cooperative approach has been developed in [Van den Bergh and Engelbrecht 2000, Van den Bergh and Engelbrecht 2001, Van den Bergh 2002] similar to the CCGA approach of Potter (refer to Section 9.6). For the cooperative PSO (CPSO), the J parameters to be optimized can be split into J swarms, where each swarm optimizes only one parameter of the problem. The optimization process within each of the J swarms occur using any of the PSO algorithms discussed previously. The difficulty with the CPSO algorithm is how to evaluate the fitness of these onedimensional particles within each swarm. The fitness of each particle of swarm Sj
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CHAPTER 16. PARTICLE SWARM
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cannot be computed in isolation from the other swarms, since a particle in a specific swarm represents just one component of the complete J-dimensional solution. A solution to this problem is to construct a context vector from the other J-l swarms by taking the global best particle from each of these J 1 swarms. The fitness of particles in swarm Sj is then calculated by replacing the j-th component in the context vector with that of the particle being evaluated. This approach to the evaluation of fitness promotes cooperation among the different swarms, since each swarm contributes to the context vector. It is important to note that the CPSO algorithm is mostly applicable to problems where the parameters to be optimized are independent of one another.
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Particle Swarm Optimization versus Evolutionary Computing and Cultural Evolution
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PSO has its roots in several disciplines, including artificial life, evolutionary computing and swarm theory. This section illustrates the similarities and differences between PSO and EC. Both paradigms are optimization algorithms which use adaptation of a population of individuals, based on natural properties. In both PSO and EC the search space is traversed using probabilistic transition rules. There are several important differences between PSO and EC. PSO has memory, while EC has no memory. Particles keep track of their best solutions, as well as that of their neighborhood. This history of best solutions plays an important role in adjusting the positions of particles. Additionally, the previous velocities are used to adjust positions. While both approaches are based on adaptation, changes are driven through learning from peers in the case of PSO, and not through genetic recombination and mutations. PSO uses no fitness function to drive the search process. Instead, the search process is guided by social interaction among peers. PSO can more closely be related to CE. In this case the population space is searched using an EP, where mutation is a function of previous mutations and distances from the best solutions. The cultural beliefs are defined by the best solutions per individual and neighborhood. In the case of more than one neighborhood, the population can be viewed as consisting of subproblems, each representing one neighborhood.
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PSO has been used mostly to find the minima and maxima of nonlinear functions [Shi and Eberhart 1999]. PSO has also been used successfully to train NNs [Eberhart et al. 1996, Kennedy and Eberhart 1999, Engelbrecht and Ismail 1999,
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16.8. CONCLUSION
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Van den Bergh 1999, Van den Bergh and Engelbrecht 2001]. In this case, each particle presents a weight vector, representing one NN. The performance of a particle is then simply the MSE over the training and test sets. A PSO has also been used successfully for human tremor analysis in order to predict Parkinson's disease [Shi and Eberhart 1999]. This section illustrates the application of PSO to find the minimum of the function /(#!, #2) = x\+x 2-, for which the minimum is at x\ = 0, x% = 0. Particles are flown in two-dimensional space. Consider the initial swarm of particles as represented by the dots in Figure 16.2. The optimum point is indicated by a cross. Figure 16.2(a) illustrates the gbest version of PSO. Particle a is the current global best solution. Initially the pbest of each individual is its current point. Therefor, only particle a influences the movement of all the particles. The arrows indicate the direction and magnitude of the change in positions. All the particles are adjusted toward particle a. The Ibest version, as illustrated in Figure 16.2(b), shows how particles are influenced by their immediate neighbors. To keep the graph readable, only some of the movements are illustrated. In neighborhood 1, both particles a and b move toward particle c, which is the best solution within that neighborhood. Considering neighborhood 2, particle d moves toward /, so does e. For the next iteration, e will be the best solution for neighborhood 2. Now d and f move toward e as illustrated in Figure 16.2(c) (only part of the solution space is illustrated). The blocks represent the previous positions. Note that e remains the best solution for neighborhood 2. Also evident is the movement toward the minimum, although slower as for gbest.
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