Structural and Multidisciplinary Optimization

, Volume 60, Issue 6, pp 2373–2389 | Cite as

A multi-objective optimization methodology based on multi-mid-range meta-models for multimodal deterministic/robust problems

  • S. KhalfallahEmail author
  • H. E. Lehtihet
Research Paper


The success of meta-model-based optimization primarily relies on how accurately the black-box functions are being represented. However, sometimes a global meta-model fails to achieve sufficient accuracy. This can be the case in multi-objective deterministic problems involving multimodal functions with competitive local optima or robust problems which require an accurate local description. This paper proposes a new methodology that deals with this type of situations and that provides the required accuracy both locally and globally. We use a set of mid-range meta-models which, in contrast to other works, are not used to construct a global meta-model but are managed both to compete and collaborate to solve the problem. They are defined across overlapping regions of interest generated by a process which resizes and moves adaptively these regions until tracking the Pareto front. The accuracy of these mid-range meta-models is also improved by a new design-of-experiment (DoE) adaptive technique allowing the suppression of some inefficient DoE points. The proposed method is implemented using standard techniques, such as non-dominated sorting genetic algorithm-II (NSGA-II), whereas the optimal shape factor of radial basis functions (RBF) is calculated by combining NSGA-II and particle swarm optimization (PSO). We also use Hager’s method to detect ill-conditioned systems and avoid propagating their outcome, which significantly improves the performance. This method is tested against difficult deterministic and robust multi-objective multimodal benchmarks and is applied to the robust optimization of an aerodynamic design case.


Meta-model-based optimization Robust design Multimodal functions CFD 



The authors would like to thank anonymous reviewers for having provided valuable criticisms and recommendations which have greatly helped improve the quality of the paper.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

Supplementary material (4.8 mb)
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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Laboratory of Turbomachinery, Ecole Militaire PolytechniqueAlgiersAlgeria
  2. 2.Department of Applied Mathematics Ecole Militaire PolytechniqueAlgiersAlgeria

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