Statistical Papers https://doi.org/10.1007/s00362-018-01075-7 REGULAR ARTICLE Approximate and exact optimal designs for 2 k factorial experiments for generalized linear models via second order cone programming Belmiro P. M. Duarte 1,2 · Guillaume Sagnol 3 Received: 19 February 2018 / Revised: 5 December 2018 © Springer-Verlag GmbH Germany, part of Springer Nature 2019 Abstract 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 nonlin- ear models when the covariates are categorical variables, i.e. factorial experiments, is scarce. We propose second order cone programming (SOCP) and Mixed Integer Sec- ond Order Programming (MISOCP) formulations to find, respectively, approximate and exact A- and D-optimal designs for 2 k factorial experiments for Generalized Lin- ear 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 det D-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. Keywords 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 B Belmiro P. M. Duarte bduarte@isec.pt Guillaume Sagnol sagnol@math.tu-berlin.de 1 Department of Chemical and Biological Engineering, Instituto Politécnico de Coimbra, Instituto Superior de Engenharia de Coimbra, Rua Pedro Nunes, Quinta da Nora, 3030-199 Coimbra, Portugal 2 CIEPQPF, Department of Chemical Engineering, University of Coimbra, Coimbra, Portugal 3 Institut für Mathematik, Technische Universität Berlin, Berlin, Germany 123