Designing efficient experiments under practical constraints is critical in both scientific research and industrial practice. Focusing on minimizing the average variance of the parameter estimates, A-optimal designs show advantages in screening factors and reducing prediction errors. Compared with other criteria, however, algorithms and software for generating A-optimal designs are scarce. In this paper, we characterize A-optimal designs under generalized linear models theoretically and develop efficient algorithms for identifying them. When a predetermined finite set of experimental settings is given, we derive analytic solutions or establish necessary and sufficient conditions for obtaining A-optimal approximate allocations. We show that a lift-one algorithm based on our formulae outperforms commonly used algorithms for finding A-optimal allocations. When continuous factors or design regions get involved, we develop a ForLion algorithm that is guaranteed to find A-optimal designs with mixed factors. Numerical studies show that our algorithms can find highly efficient designs with reduced numbers of distinct experimental settings, which may save both experimental time and cost significantly. Along with a rounding-off algorithm that converts approximate allocations to exact ones, we demonstrate that stratified samplers based on A-optimal allocations may provide more accurate parameter estimates than commonly used samplers.

翻译：在实际约束条件下设计高效实验，在科学研究和工业实践中均至关重要。A-最优设计通过最小化参数估计的平均方差，在因子筛选与预测误差降低方面展现出优势。然而与其他准则相比，生成A-最优设计的算法与软件工具仍较为匮乏。本文从理论上刻画了广义线性模型下的A-最优设计，并提出了高效辨识算法。当给定预设的有限实验设置集合时，我们推导出解析解或建立充要条件以获得A-最优近似分配。研究表明，基于本文公式的升一算法在寻找A-最优分配方面优于常用算法。针对涉及连续因子或设计区域的情形，我们开发了保证能获取含混合因子A-最优设计的ForLion算法。数值实验表明，本文算法能以较少的互异实验设置数量获得高效设计，从而显著节省实验时间与成本。结合将近似分配转化为精确分配的舍入算法，我们证实基于A-最优分配的分层抽样器相较于常用抽样器能提供更精确的参数估计。