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A very simple approach to cope with constraints is to reject infeasible particles, as follows: Do not allow infeasible particles to be selected as personal best or neighborhood global best solutions In doing so, infeasible particles will never in uence other particles in the swarm Infeasible particles are, however, pulled back to feasible space due to the fact that personal best and neighborhood best positions are in feasible space This approach can only be e cient if the ratio of number of infeasible particles to feasible particles is small If this ratio is too large, the swarm may not have enough diversity to e ectively cover the (feasible) space Reject infeasible particles by replacing them with new randomly generated positions from the feasible space Reinitialization within feasible space gives these particles an opportunity to become best solutions However, it is also possible that these particles (or any other particle for that matter), may roam outside the boundaries of feasible space This approach may be bene cial in cases where the feasible space is made up of a number of disjointed regions The strategy will allow particles to cross boundaries of one feasible region to explore another feasible region Most applications of PSO to constrained problems make use of penalty methods to penalize those particles that violate constraints [663, 838, 951] Similar to the approach discussed in Section 1261, Shi and Krohling [784] and Laskari et al [504] converted the constrained problem to an unconstrained Lagrangian Repair methods, which allow particles to roam into infeasible space, have also been developed These are simple methods that apply repairing operators to change infeasible particles to represent feasible solutions Hu and Eberhart et al [388] developed an approach where particles are not allowed to be attracted by infeasible particles The personal best of a particle changes only if the tness of the current position is better and if the current position violates no constraints This ensures that the personal best positions are always feasible Assuming that neighborhood best positions are selected from personal best positions, it is also guaranteed that neighborhood best positions are feasible In the case where the neighborhood best positions are selected from the current particle positions, neighborhood best positions should be updated only if no constraint is violated Starting with a feasible set of particles, if a particle moves out of feasible space, that particle will be pulled back towards feasible space Allowing particles to roam into infeasible space, and repairing them over time by.

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moving back to feasible best positions facilitates better exploration If the feasible space consists of disjointed feasible regions, the chances are increased for particles to explore di erent feasible regions El-Gallad et al [234] replaced infeasible particles with their feasible personal best positions This approach assumes feasible initial particles and that personal best positions are replaced only with feasible solutions The approach is very similar to that of Hu and Eberhart [388], but with less diversity Replacement of an infeasible particle with its feasible personal best, forces an immediate repair The particle is immediately brought back into feasible space The approach of Hu and Eberhart allow an infeasible particle to explore more by pulling it back into feasible space over time But, keep in mind that during this exploration, the infeasible particle will have no in uence on the rest of the swarm while it moves within infeasible space Venter and Sobieszczanski-Sobieski [874, 875] proposed repair of infeasible solutions by setting vi (t) = 0 vi (t + 1) = c1 r1 (t)(yi (t) xi (t)) + c2 r2 (t)( (t) xi (t)) y (1691) (1692)

for all infeasible particles, i In other words, the memory of previous velocity (direction of movement) is deleted for infeasible particles, and the new velocity depends only on the cognitive and social components Removal of the momentum has the e ect that infeasible particles are pulled back towards feasible space (assuming that the personal best positions are only updated if no constraints are violated)

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