Particle swarm optimization (analysis) - Revision history

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=== 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<nowiki>'</nowiki>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) + ''C''1''r''1(''xpbest − xi''(''t'')) + ''C''2''r''2(''xgbest − xi''(''t'')),
|}
where ''W'' is the inertia weight, ''C''1 and ''C''2 are the learning factors (usually defined as constants), and ''r''1, ''r''2 ∈ <nowiki>[</nowiki>0,1<nowiki>]</nowiki> 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.

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