UCLA

We use semidefinite programming (SDP) to find a variety of optimal designs for multi-response linearmodels with multiple factors, and for the first time, extend the methodology to find optimal designs formulti-response nonlinear models and generalized linear models with multiple factors. We construct transformationsthat (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 multiresponsemodels with multiple factors and (iii) correct erroneously reported optimal designs in the literaturecaused by formulation issues.We also derive invariance properties of optimal designs and their dependenceon the covariance matrix of the correlated errors, which are helpful for reducing the computation timefor finding optimal designs. Our applications include finding A-, As-, c-, and D-optimal designs for multiresponsemulti-factor polynomial models, locally c- and D-optimal designs for a bivariate Emax responsemodel and for a bivariate Probit model useful in the biosciences.