### Abstract

Ideally, the penalization coefficients should be adaptive. While research on adaptive coefficients is extensive in the literature, 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. and by Muñoz Zavala et al. 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.

Original language | English |
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Title of host publication | Proceedings of the Tenth International Conference on Computational Structures Technology |

Editors | B.H.V. Topping, J.M. Adam, F.J. Pallarés, R. Bru, M.L. Romero |

Place of Publication | Stirlingshire |

Publisher | Saxe-Coburg Publications |

ISBN (Print) | 978-1-905088-38-6 |

DOIs | |

Publication status | Published - 2010 |

Externally published | Yes |

Event | 10th Int. Conference on Computational Structures Technology - Universidad Politécnica de Valencia, Valencia, Spain Duration: 14 Sep 2010 → 17 Sep 2010 Conference number: 10 http://www.civil-comp.com/conf/cst2010.htm |

### Publication series

Name | Civil-Comp Proceedings |
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ISSN (Electronic) | 1759-3433 |

### Conference

Conference | 10th Int. Conference on Computational Structures Technology |
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Country | Spain |

City | Valencia |

Period | 14/09/10 → 17/09/10 |

Internet address |

### Fingerprint

### Keywords

- particle swarm optimization
- pseudo-adaptive tolerance relaxation
- penalization method
- constant coefficients

### Cite this

*Proceedings of the Tenth International Conference on Computational Structures Technology*[123] (Civil-Comp Proceedings). Stirlingshire: Saxe-Coburg Publications. https://doi.org/10.4203/ccp.93.123

**Pseudo-Adaptive Penalization to Handle Constraints in Particle Swarm Optimizers.** / Innocente, Mauro; Sienz, Johann.

Research output: Chapter in Book/Report/Conference proceeding › Conference proceeding

*Proceedings of the Tenth International Conference on Computational Structures Technology.*, 123, Civil-Comp Proceedings, Saxe-Coburg Publications, Stirlingshire, 10th Int. Conference on Computational Structures Technology, Valencia, Spain, 14/09/10. https://doi.org/10.4203/ccp.93.123

}

TY - GEN

T1 - Pseudo-Adaptive Penalization to Handle Constraints in Particle Swarm Optimizers

AU - Innocente, Mauro

AU - Sienz, Johann

PY - 2010

Y1 - 2010

N2 - 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, 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. and by Muñoz Zavala et al. 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.

AB - 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, 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. and by Muñoz Zavala et al. 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.

KW - particle swarm optimization

KW - pseudo-adaptive tolerance relaxation

KW - penalization method

KW - constant coefficients

U2 - 10.4203/ccp.93.123

DO - 10.4203/ccp.93.123

M3 - Conference proceeding

SN - 978-1-905088-38-6

T3 - Civil-Comp Proceedings

BT - Proceedings of the Tenth International Conference on Computational Structures Technology

A2 - Topping, B.H.V.

A2 - Adam, J.M.

A2 - Pallarés, F.J.

A2 - Bru, R.

A2 - Romero, M.L.

PB - Saxe-Coburg Publications

CY - Stirlingshire

ER -