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_<i>s</i>-, c-, and D-optimal designs for multi-response multi-factor polynomial models, locally c- and D-optimal designs for a bivariate Emax response model and for a bivariate Probit model useful in the biosciences.我们采用半定规划（semidefinite programming, SDP），针对多因素多响应线性模型求解各类最优设计，并首次将该方法拓展至多因素多响应非线性模型与广义线性模型的最优设计求解中。我们构建了三类变换：其一，优化最优设计问题向半定规划问题的建模形式；其二，支持将半定规划方法从线性模型拓展至多因素多响应非线性模型的最优设计求解；其三，修正了现有文献中因建模形式缺陷导致的错误报道的最优设计结果。此外，我们推导了最优设计的不变性性质，以及其与相关误差协方差矩阵的依赖关系，这有助于缩短最优设计求解的计算时长。我们的应用场景包括：针对多因素多响应多项式模型求解A-、Aₛ-、c-与D-最优设计；针对生物科学领域常用的双变量Emax响应模型与双变量概率单位（Probit）模型，求解局部c-最优设计与局部D-最优设计。