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One-dimensional nested maximin designs

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  • Edwin Dam
  • Bart Husslage
  • Dick Hertog

Abstract

The design of computer experiments is an important step in black-box evaluation and optimization processes. When dealing with multiple black-box functions the need often arises to construct designs for all black boxes jointly, instead of individually. These so-called nested designs are particularly useful as training and test sets for fitting and validating metamodels, respectively. Furthermore, nested designs can be used to deal with linking parameters and sequential evaluations. In this paper, we introduce one-dimensional nested maximin designs. We show how to nest two designs optimally and develop a heuristic to nest three and four designs. These nested maximin designs can be downloaded from the website http://www.spacefillingdesigns.nl. Furthermore, it is proven that the loss in space-fillingness, with respect to traditional maximin designs, is at most 14.64 and 19.21%, when nesting two and three designs, respectively.
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Suggested Citation

  • Edwin Dam & Bart Husslage & Dick Hertog, 2010. "One-dimensional nested maximin designs," Journal of Global Optimization, Springer, vol. 46(2), pages 287-306, February.
    • Handle: RePEc:spr:jglopt:v:46:y:2010:i:2:p:287-306
    DOI: 10.1007/s10898-009-9426-y
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    1. J P C Kleijnen & W C M van Beers, 2004. "Application-driven sequential designs for simulation experiments: Kriging metamodelling," Journal of the Operational Research Society, Palgrave Macmillan;The OR Society, vol. 55(8), pages 876-883, August.
    2. den Hertog, Dick & Stehouwer, Peter, 2002. "Optimizing color picture tubes by high-cost nonlinear programming," European Journal of Operational Research, Elsevier, vol. 140(2), pages 197-211, July.
    3. Edwin R. van Dam & Gijs Rennen & Bart Husslage, 2009. "Bounds for Maximin Latin Hypercube Designs," Operations Research, INFORMS, vol. 57(3), pages 595-608, June.
    4. Rennen, G. & Husslage, B.G.M. & van Dam, E.R. & den Hertog, D., 2009. "Nested Maximin Latin Hypercube Designs," Discussion Paper 2009-06, Tilburg University, Center for Economic Research.
    5. Edwin R. van Dam & Bart Husslage & Dick den Hertog & Hans Melissen, 2007. "Maximin Latin Hypercube Designs in Two Dimensions," Operations Research, INFORMS, vol. 55(1), pages 158-169, February.
    6. Husslage, B.G.M. & van Dam, E.R. & den Hertog, D. & Stehouwer, H.P. & Stinstra, E., 2003. "Collaborative metamodelling : Coordinating simulation-based product design," Other publications TiSEM 0196e58f-78a8-4653-a48b-8, Tilburg University, School of Economics and Management.
    7. Chao-Ton Su & Mu-Chen Chen & Hsiao-Ling Chan, 2005. "Applying neural network and scatter search to optimize parameter design with dynamic characteristics," Journal of the Operational Research Society, Palgrave Macmillan;The OR Society, vol. 56(10), pages 1132-1140, October.
    8. van Dam, E.R. & Husslage, B.G.M. & den Hertog, D. & Melissen, H., 2005. "Maximin Latin Hypercube Designs in Two Dimensions," Other publications TiSEM 288828ce-b56b-41d8-9903-1, Tilburg University, School of Economics and Management.
    9. Rennen, G. & Husslage, B.G.M. & van Dam, E.R. & den Hertog, D., 2009. "Nested Maximin Latin Hypercube Designs," Other publications TiSEM 1c504ec0-f357-42d2-9c92-9, Tilburg University, School of Economics and Management.
    10. Castillo, Ignacio & Kampas, Frank J. & Pintér, János D., 2008. "Solving circle packing problems by global optimization: Numerical results and industrial applications," European Journal of Operational Research, Elsevier, vol. 191(3), pages 786-802, December.
    11. van Dam, E.R. & Rennen, G. & Husslage, B.G.M., 2007. "Bounds for Maximin Latin Hypercube Designs," Other publications TiSEM da0c15be-f18e-474e-b557-f, Tilburg University, School of Economics and Management.
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    Cited by:

    1. Rennen, G. & Husslage, B.G.M. & van Dam, E.R. & den Hertog, D., 2009. "Nested Maximin Latin Hypercube Designs," Discussion Paper 2009-06, Tilburg University, Center for Economic Research.
    2. Husslage, B.G.M. & van Dam, E.R. & den Hertog, D., 2005. "Nested Maximin Latin Hypercube Designs in Two Dimensions," Discussion Paper 2005-79, Tilburg University, Center for Economic Research.
    3. Husslage, B.G.M. & van Dam, E.R. & den Hertog, D., 2005. "Nested Maximin Latin Hypercube Designs in Two Dimensions," Other publications TiSEM 3e013144-3e4c-460c-96bc-1, Tilburg University, School of Economics and Management.
    4. Grishagin, Vladimir & Israfilov, Ruslan & Sergeyev, Yaroslav, 2018. "Convergence conditions and numerical comparison of global optimization methods based on dimensionality reduction schemes," Applied Mathematics and Computation, Elsevier, vol. 318(C), pages 270-280.
    5. Hao Chen & Yan Zhang & Xue Yang, 2021. "Uniform projection nested Latin hypercube designs," Statistical Papers, Springer, vol. 62(4), pages 2031-2045, August.
    6. János Pintér & Zoltán Horváth, 2013. "Integrated experimental design and nonlinear optimization to handle computationally expensive models under resource constraints," Journal of Global Optimization, Springer, vol. 57(1), pages 191-215, September.
    7. Vieira Jr., Hélcio & Sanchez, Susan & Kienitz, Karl Heinz & Belderrain, Mischel Carmen Neyra, 2011. "Generating and improving orthogonal designs by using mixed integer programming," European Journal of Operational Research, Elsevier, vol. 215(3), pages 629-638, December.

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    More about this item

    Keywords

    Computer simulation; Global optimization; Linking parameter; Maximin design; Mixed integer linear programming; Packing problem; Space-filling; Training and test set;
    All these keywords.

    JEL classification:

    • C90 - Mathematical and Quantitative Methods - - Design of Experiments - - - General

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