  Despite the increasing prevalence of vector observations, computation of
optimal experimental design for multi-response models has received limited
attention. To address this problem within the framework of approximate designs,
we introduce mREX, an algorithm that generalizes the randomized exchange
algorithm REX (J Am Stat Assoc 115:529, 2020), originally specialized for
single-response models. The mREX algorithm incorporates several improvements: a
novel method for computing efficient sparse initial designs, an extension to
all differentiable Kiefer's optimality criteria, and an efficient method for
performing optimal exchanges of weights. For the most commonly used
D-optimality criterion, we propose a technique for optimal weight exchanges
based on the characteristic matrix polynomial. The mREX algorithm is applicable
to linear, nonlinear, and generalized linear models, and scales well to large
problems. It typically converges to optimal designs faster than available
alternative methods, although it does not require advanced mathematical
programming solvers. We demonstrate the application of mREX to bivariate
dose-response Emax models for clinical trials, both without and with the
inclusion of covariates.
