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 to find A-optimal designs with mixed factors. Numerical studies on real-data experiments 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 A-optimal designs may provide more accurate parameter estimates than commonly used alternatives.

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