Difference between revisions of "Particle swarm optimization (analysis)"
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Revision as of 16:28, 4 April 2013
Multi-objective particle swarm optimization (MOPSO)
The particle swarm optimization (PSO) algorithm, initially proposed by Kennedy and Eberhart1, is a direct search algorithm based on the simulation of the social behavior of birds within a flock. The swarm is typically modeled by particles in the multidimensional space that have a position and a velocity. These particles fly through hyperspace and have two essential reasoning capabilities: their memory of their own best position and knowledge of the global or their neighborhood's best.
Let xi(t) denote the position of particle pi at time step t. The position of pi is then changed by adding a velocity vi(t) to its current position, i.e.
| xi(t) = xi(t − 1) + vi(t). |
Let gbest is the position of the best particle from the entire swarm and pbest is the position of the neighborhood best that the particle obtained by communicating with a subset of the swarm. In this case, velocity equation is given by
| vi(t) = W·vi(t − 1) + C1r1(xpbest − xi(t)) + C2r2(xgbest − xi(t)), |
where W is the inertia weight, C1 and C2 are the learning factors (usually defined as constants), and r1, r2 ∈ [0,1] are random values.
Here we used a multiple-objective particle swarm optimizers based on the paper of Sierra and Coello2. This optimizer allows to take into account parametric constraints and uses a crowding factor for the leaders selection as well as the following mutation operators: an uniform mutation operator, where the variability range allowed for each decision variable is constant over generations, and a non-uniform mutation operator, where such range decreases over time. These operators modify the values of the particle decision variables with a certain probability.
References
- J Kennedy, RC Eberhart. "Particle Swarm Optimization". Proceedings of the 1995 IEEE International Conference on Neural Networks, Piscataway, New Jersey, IEEE Service Center, 1995, 1942-1948.
- MR Sierra and CA Coello Coello "Improving PSO-Based Multi-objective Optimization Using Crowding, Mutation and e-Dominance." Evolutionary Multi-Criterion Optimization, 3410/2005: 505-519, Springer Berlin/Heidelberg.
