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.

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