Approximate and exact optimal designs for \(2^k\) factorial experiments for generalized linear models via second order cone programming

  • Belmiro P. M. DuarteEmail author
  • Guillaume Sagnol
Regular Article


Model-based optimal designs of experiments (M-bODE) for nonlinear models are typically hard to compute. The literature on the computation of M-bODE for nonlinear models when the covariates are categorical variables, i.e. factorial experiments, is scarce. We propose second order cone programming (SOCP) and Mixed Integer Second Order Programming (MISOCP) formulations to find, respectively, approximate and exact A- and D-optimal designs for \(2^k\) factorial experiments for Generalized Linear Models (GLMs). First, locally optimal (approximate and exact) designs for GLMs are addressed using the formulation of Sagnol (J Stat Plan Inference 141(5):1684–1708, 2011). Next, we consider the scenario where the parameters are uncertain, and new formulations are proposed to find Bayesian optimal designs using the A- and log detD-optimality criteria. A quasi Monte-Carlo sampling procedure based on the Hammersley sequence is used for computing the expectation in the parametric region of interest. We demonstrate the application of the algorithm with the logistic, probit and complementary log–log models and consider full and fractional factorial designs.


D-optimal designs \(2^k\) Factorial experiments Exact designs Second order cone programming Generalized linear models Quasi-Monte Carlo sampling 

Mathematics Subject Classification

62K05 90C47 



The authors thank Radoslav Harman of Comenius University in Bratislava for valuable comments and advice on an earlier draft of the manuscript. We also thank two anonymous reviewers whose comments allowed to undoubtedly improving the quality of the paper.


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Copyright information

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

Authors and Affiliations

  1. 1.Department of Chemical and Biological EngineeringInstituto Politécnico de Coimbra, Instituto Superior de Engenharia de CoimbraCoimbraPortugal
  2. 2.CIEPQPF, Department of Chemical EngineeringUniversity of CoimbraCoimbraPortugal
  3. 3.Institut für MathematikTechnische Universität BerlinBerlinGermany

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