Optimal designs for multi-response nonlinear regression models with several factors via semi-definite programming

We use semi-definite 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 <i>E</i>_<i>max</i> response model and for a bivariate Probit model useful in the biosciences.

本研究采用半正定规划（semi-definite programming, SDP），针对多因子多响应线性模型求解多类最优设计，并首次将该方法拓展至多因子多响应非线性模型与广义线性模型的最优设计求解工作中。本研究构建了若干变换，其一可优化最优设计问题向半正定规划问题的转化形式，其二可将半正定规划方法的适用范围从线性模型拓展至多因子多响应非线性模型，其三可修正已有文献中因建模形式不当而错误报道的最优设计结果。此外，本研究推导了最优设计的不变性性质及其与相关误差协方差矩阵的依赖关系，该结论可有效缩短最优设计求解的计算耗时。本研究的应用场景包括：针对多因子多响应多项式模型求解A-、A_s-、c-及D-最优设计；针对双变量E_max响应模型以及生物科学领域常用的双变量概率单位模型（Probit）求解局部c-与D-最优设计。