Keywords: particle swarm optimization, pseudo-adaptive tolerance relaxation, penalization method, constant coefficients.

Since particle swarm optimization is suitable for unconstrained problems only, some external technique needs to be incorporated to deal with constrained problems. One of the most popular techniques is the penalization method, where infeasible solutions are penalized by increasing the objective function value in minimization problems. The key is in the amount of penalization, which is typically linked to the amount of constraint violation. By turning the constrained problem into an unconstrained one, these methods are well suited for particle swarm optimizers because they do not disrupt the normal dynamics of the swarm. However they present the downside that problem-dependent penalization coefficients are involved: an excessive penalization might lead to premature convergence, whereas too mild a penalization might lead to infeasible solutions being chosen over feasible ones.

Ideally, the penalization coefficients should be adaptive. While research on adaptive coefficients is extensive in the literature [1,3], a different adaptive scheme is proposed here where the coefficients are kept constant. The procedure consists of an initial self-tuned relaxation of the constraint violation tolerances, followed by a pseudo-adaptive decrease of the relaxations. The self-tuning is performed so that an approximate target feasibility ratio is reached. The pseudo-adaptive decrease is linked to the number of potential feasible solutions found at the current time-step. Thus, by linking the penalization to the constraint violations beyond the pseudo-adaptive tolerance rather than to the actual constraint violations, a pseudo-adaptive penalization is achieved.

A particle swarm optimizer equipped with this constraint-handling mechanism is successfully tested on a suite of thirteen constrained problems. For comparison, the experiments are also performed without tolerance relaxations, and with the initial self-tuned relaxation followed by a deterministic decrease. Comparisons to the results reported by Toscano Pulido et al. [4] and by Muñoz Zavala et al. [2] are offered as frames of reference. The pseudo-adaptive tolerance relaxations scheme is successful in improving the solutions obtained for problems with low feasibility ratios and/or whose solutions are near or on the boundaries.

1

C.A. Coello Coello, "Use of a self-adaptive penalty approach for engineering optimization problems", Computers in Industry, 41, 113-127, 2000. doi:10.1016/S0166-3615(99)00046-9

2

A.E. Muñoz Zavala, A. Hernández Aguirre, E.R. Villa Diharce, "Constrained Optimization via Particle Evolutionary Swarm Optimization Algorithm (PESO)", Proceedings of the 2005 Genetic and Evolutionary Computation Conference (GECCO'05), 209-216, Washington, DC, 2005. doi:10.1145/1068009.1068041

3

K.E. Parsopoulos, M.N. Vrahatis, "Particle Swarm Optimization Method for Constrained Optimization Problems", Proceedings of the Euro-International Symposium on Computational Intelligence, 2002.

4

G. Toscano Pulido, C.A. Coello Coello, "A Constraint-Handling Mechanism for Particle Swarm Optimization", Proceedings of the IEEE Congress on Evolutionary Computation, 2, 1396-1403, Portland, 2004. doi:10.1109/CEC.2004.1331060

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