Structural and Multidisciplinary Optimization

, Volume 45, Issue 3, pp 417–431 | Cite as

Novel insights for multi-objective optimisation in engineering using Normal Boundary Intersection and (Enhanced) Normalised Normal Constraint

Research Paper
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Abstract

Normal Boundary Intersection (NBI) and (Enhanced) Normalised Normal Constraint (E)NNC are attractive and popular approaches to generate an approximation of the Pareto set in nonlinear multi-objective optimisation problems. All three methods are based on similar ideas, but do not always yield identical results, which may confuse practitioners. Hence, the current paper provides theoretical insights in the conditions under which identical results are obtained. Typically, NBI and ENNC are able to generate the same candidate Pareto points, if all additional inequalities in the ENNC subproblem are active. In general, NBI and NNC do not return the same points when three or more objectives are considered. Equivalence relations between the resulting lagrange multipliers for the additional NBI and ENNC (in)equality constraints have been derived. Moreover, the obtained relations have lead to novel criteria for detecting non-Pareto optimal points that in adverse situations maybe generated by these methods. The major advantage is that the removal criteria do not rely on a time-consuming pairwise comparison but only need matrix multiplications. A Matlab implementation has been added for completeness. The insights are illustrated for a general nonlinear bi-objective and three-objective optimisation problem, and a dynamic three-objective tubular reactor optimisation problem from chemical engineering. Finally, practical guidelines are added.

Keywords

Multiobjective optimization Optimal control (Enhanced) Normalized Normal Constraint Normal Boundary Intersection 
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Notes

Acknowledgments

Work supported in part by Projects OT/10/035, OPTEC (Center-of-Excellence Optimization in Engineering) PFV/10/002 and SCORES4CHEM KP/09/005 of the Katholieke Universiteit Leuven, and by the Belgian Program on Interuniversity Poles of Attraction, initiated by the Belgian Federal Science Policy Office. J.F. Van Impe holds the chair Safety Engineering sponsored by the Belgian chemistry and life sciences federation essenscia.

Supplementary material

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Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.Chemical and Biochemical Process Technology and Control (BioTeC) & Optimization in Engineering Center (OPTEC), Department of Chemical EngineeringK.U. LeuvenLeuvenBelgium

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