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On the choice of the low-dimensional domain for global optimization via random embeddings

Author

Listed:
  • Mickaël Binois

    (The University of Chicago Booth School of Business)

  • David Ginsbourger

    (Idiap Research Institute, Centre du Parc
    University of Bern)

  • Olivier Roustant

    (LIMOS)

Abstract

The challenge of taking many variables into account in optimization problems may be overcome under the hypothesis of low effective dimensionality. Then, the search of solutions can be reduced to the random embedding of a low dimensional space into the original one, resulting in a more manageable optimization problem. Specifically, in the case of time consuming black-box functions and when the budget of evaluations is severely limited, global optimization with random embeddings appears as a sound alternative to random search. Yet, in the case of box constraints on the native variables, defining suitable bounds on a low dimensional domain appears to be complex. Indeed, a small search domain does not guarantee to find a solution even under restrictive hypotheses about the function, while a larger one may slow down convergence dramatically. Here we tackle the issue of low-dimensional domain selection based on a detailed study of the properties of the random embedding, giving insight on the aforementioned difficulties. In particular, we describe a minimal low-dimensional set in correspondence with the embedded search space. We additionally show that an alternative equivalent embedding procedure yields simultaneously a simpler definition of the low-dimensional minimal set and better properties in practice. Finally, the performance and robustness gains of the proposed enhancements for Bayesian optimization are illustrated on numerical examples.

Suggested Citation

  • Mickaël Binois & David Ginsbourger & Olivier Roustant, 2020. "On the choice of the low-dimensional domain for global optimization via random embeddings," Journal of Global Optimization, Springer, vol. 76(1), pages 69-90, January.
  • Handle: RePEc:spr:jglopt:v:76:y:2020:i:1:d:10.1007_s10898-019-00839-1
    DOI: 10.1007/s10898-019-00839-1
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    References listed on IDEAS

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    1. D. Huang & T. Allen & W. Notz & N. Zeng, 2006. "Global Optimization of Stochastic Black-Box Systems via Sequential Kriging Meta-Models," Journal of Global Optimization, Springer, vol. 34(3), pages 441-466, March.
    2. Tipaluck Krityakierne & Taimoor Akhtar & Christine A. Shoemaker, 2016. "SOP: parallel surrogate global optimization with Pareto center selection for computationally expensive single objective problems," Journal of Global Optimization, Springer, vol. 66(3), pages 417-437, November.
    3. Mishra, SK, 2006. "Global Optimization by Differential Evolution and Particle Swarm Methods: Evaluation on Some Benchmark Functions," MPRA Paper 1005, University Library of Munich, Germany.
    4. Roustant, Olivier & Ginsbourger, David & Deville, Yves, 2012. "DiceKriging, DiceOptim: Two R Packages for the Analysis of Computer Experiments by Kriging-Based Metamodeling and Optimization," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 51(i01).
    5. Nicolas Courrier & Pierre-Alain Boucard & Bruno Soulier, 2016. "Variable-fidelity modeling of structural analysis of assemblies," Journal of Global Optimization, Springer, vol. 64(3), pages 577-613, March.
    6. Mishra, SK, 2006. "Performance of Differential Evolution and Particle Swarm Methods on Some Relatively Harder Multi-modal Benchmark Functions," MPRA Paper 449, University Library of Munich, Germany.
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    Cited by:

    1. Dawei Zhan & Huanlai Xing, 2020. "Expected improvement for expensive optimization: a review," Journal of Global Optimization, Springer, vol. 78(3), pages 507-544, November.

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