We study approximate K-optimal designs for various regression models by minimizing the condition number of the information matrix. This minimizes the error sensitivity in the computation of the least squares estimator of regression parameters and also avoids the multicollinearity in regression. Using matrix and optimization theory, we derive several theoretical results of K-optimal designs, including convexity of K-optimality criterion, lower bounds of the condition number, and symmetry properties of K-optimal designs. A general numerical method is developed to find K-optimal designs for any regression model on a discrete design space. In addition, specific results are obtained for polynomial, trigonometric and second-order response models.

我们通过最小化信息矩阵的条件数来研究各种回归模型的近似 K 最优设计。这最大限度地降低了回归参数最小二乘估计量计算中的误差敏感性，也避免了回归中的多重共线性。利用矩阵和优化理论，我们得出了 K 最优设计的几个理论结果，包括 K 最优准则的凸性、条件数的下界和 K 最优设计的对称性。开发了一种通用数值方法来为离散设计空间上的任何回归模型找到 K 最优设计。此外，还针对多项式、三角函数和二阶响应模型获得了具体结果。