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A Semi-Infinite Programming based algorithm for determining T-optimum designs for model discrimination

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  • Duarte, Belmiro P.M.
  • Wong, Weng Kee
  • Atkinson, Anthony C.

Abstract

T-optimum designs for model discrimination are notoriously difficult to find because of the computational difficulty involved in solving an optimization problem that involves two layers of optimization. Only a handful of analytical T-optimal designs are available for the simplest problems; the rest in the literature are found using specialized numerical procedures for a specific problem. We propose a potentially more systematic and general way for finding T-optimal designs using a Semi-Infinite Programming (SIP) approach. The strategy requires that we first reformulate the original minimax or maximin optimization problem into an equivalent semi-infinite program and solve it using an exchange-based method where lower and upper bounds produced by solving the outer and the inner programs, are iterated to convergence. A global Nonlinear Programming (NLP) solver is used to handle the subproblems, thus finding the optimal design and the least favorable parametric configuration that minimizes the residual sum of squares from the alternative or test models. We also use a nonlinear program to check the global optimality of the SIP-generated design and automate the construction of globally optimal designs. The algorithm is successfully used to produce results that coincide with several T-optimal designs reported in the literature for various types of model discrimination problems with normally distributed errors. However, our method is more general, merely requiring that the parameters of the model be estimated by a numerical optimization.

Suggested Citation

  • Duarte, Belmiro P.M. & Wong, Weng Kee & Atkinson, Anthony C., 2015. "A Semi-Infinite Programming based algorithm for determining T-optimum designs for model discrimination," Journal of Multivariate Analysis, Elsevier, vol. 135(C), pages 11-24.
    • Handle: RePEc:eee:jmvana:v:135:y:2015:i:c:p:11-24
    DOI: 10.1016/j.jmva.2014.11.006
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    References listed on IDEAS

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    1. J. López‐Fidalgo & C. Tommasi & P. C. Trandafir, 2007. "An optimal experimental design criterion for discriminating between non‐normal models," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 69(2), pages 231-242, April.
    2. Douglas P. Wiens, 2009. "Robust discrimination designs," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 71(4), pages 805-829, September.
    3. Lopez, Marco & Still, Georg, 2007. "Semi-infinite programming," European Journal of Operational Research, Elsevier, vol. 180(2), pages 491-518, July.
    4. Dariusz Uciński & Barbara Bogacka, 2005. "T‐optimum designs for discrimination between two multiresponse dynamic models," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 67(1), pages 3-18, February.
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    1. Duarte, Belmiro P.M. & Sagnol, Guillaume & Wong, Weng Kee, 2018. "An algorithm based on semidefinite programming for finding minimax optimal designs," Computational Statistics & Data Analysis, Elsevier, vol. 119(C), pages 99-117.
    2. Chiara Tommasi & Juan M. Rodríguez-Díaz & Jesús F. López-Fidalgo, 2023. "An equivalence theorem for design optimality with respect to a multi-objective criterion," Statistical Papers, Springer, vol. 64(4), pages 1041-1056, August.
    3. Lucy L. Gao & Julie Zhou, 2017. "D-optimal designs based on the second-order least squares estimator," Statistical Papers, Springer, vol. 58(1), pages 77-94, March.
    4. David Mogalle & Philipp Seufert & Jan Schwientek & Michael Bortz & Karl-Heinz Küfer, 2024. "Computing T-optimal designs via nested semi-infinite programming and twofold adaptive discretization," Computational Statistics, Springer, vol. 39(5), pages 2451-2478, July.
    5. Weng Kee Wong & Yue Yin & Julie Zhou, 2019. "Using SeDuMi to find various optimal designs for regression models," Statistical Papers, Springer, vol. 60(5), pages 1583-1603, October.
    6. Duarte, Belmiro P.M. & Atkinson, Anthony C. & Granjo, Jose F.O & Oliveira, Nuno M.C, 2022. "Optimal design of experiments for implicit models," LSE Research Online Documents on Economics 107584, London School of Economics and Political Science, LSE Library.
    7. Belmiro P. M. Duarte, 2023. "Exact Optimal Designs of Experiments for Factorial Models via Mixed-Integer Semidefinite Programming," Mathematics, MDPI, vol. 11(4), pages 1-17, February.

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