Optimal Designs for Multi-Response Nonlinear Regression Models With Several Factors via Semidefinite Programming

We use semidefinite programming (SDP) to find a variety of optimal designs for multi-response linear models with multiple factors, and for the first time, extend the methodology to find optimal designs for multi-response nonlinear models and generalized linear models with multiple factors. We construct transformations that (i) facilitate improved formulation of the optimal design problems into SDP problems, (ii) enable us to extend SDP methodology to find optimal designs from linear models to nonlinear multi-response models with multiple factors and (iii) correct erroneously reported optimal designs in the literature caused by formulation issues. We also derive invariance properties of optimal designs and their dependence on the covariance matrix of the correlated errors, which are helpful for reducing the computation time for finding optimal designs. Our applications include finding A-, A_s-, c-, and D-optimal designs for multi-response multi-factor polynomial models, locally c- and D-optimal designs for a bivariate $E_{}$ response model and for a bivariate Probit model useful in the biosciences.