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Fast calculation of multiobjective probability of improvement and expected improvement criteria for Pareto optimization

Abstract

The use of surrogate based optimization (SBO) is widely spread in engineering design to reduce the number of computational expensive simulations. However, “real-world” problems often consist of multiple, conflicting objectives leading to a set of competitive solutions (the Pareto front). The objectives are often aggregated into a single cost function to reduce the computational cost, though a better approach is to use multiobjective optimization methods to directly identify a set of Pareto-optimal solutions, which can be used by the designer to make more efficient design decisions (instead of weighting and aggregating the costs upfront). Most of the work in multiobjective optimization is focused on multiobjective evolutionary algorithms (MOEAs). While MOEAs are well-suited to handle large, intractable design spaces, they typically require thousands of expensive simulations, which is prohibitively expensive for the problems under study. Therefore, the use of surrogate models in multiobjective optimization, denoted as multiobjective surrogate-based optimization, may prove to be even more worthwhile than SBO methods to expedite the optimization of computational expensive systems. In this paper, the authors propose the efficient multiobjective optimization (EMO) algorithm which uses Kriging models and multiobjective versions of the probability of improvement and expected improvement criteria to identify the Pareto front with a minimal number of expensive simulations. The EMO algorithm is applied on multiple standard benchmark problems and compared against the well-known NSGA-II, SPEA2 and SMS-EMOA multiobjective optimization methods.

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Notes

  1. 1.

    Improving the overall accuracy of the surrogate model (space-filling).

  2. 2.

    Enhancing the accuracy of the surrogate model solely in the region of the (current) optimum.

  3. 3.

    Improving or augmenting the Pareto front.

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Acknowledgments

This work was supported by the Fund for Scientific Research in Flanders (FWO-Vlaanderen). Ivo Couckuyt and Dirk Deschrijver are post-doctoral research fellows of FWO-Vlaanderen. This research has (partially) been funded by the Interuniversity Attraction Poles Program BESTCOM initiated by the Belgian Science Policy Office.

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Correspondence to Ivo Couckuyt.

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Couckuyt, I., Deschrijver, D. & Dhaene, T. Fast calculation of multiobjective probability of improvement and expected improvement criteria for Pareto optimization. J Glob Optim 60, 575–594 (2014). https://doi.org/10.1007/s10898-013-0118-2

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Keywords

  • Multiobjective optimization
  • Expected improvement
  • Probability of improvement
  • Hypervolume
  • Kriging
  • Gaussian process