Ascent with Quadratic Assistance for the Construction of Exact Experimental Designs

In the area of statistical planning, there is a large body of theoretical knowledge and computational experience concerning so-called optimal approximate designs of experiments. However, for an approximate design to be executed in practice, it must be converted into an exact, i.e., integer, design, which is usually done via rounding procedures. Although rapid, rounding procedures have many drawbacks; in particular, they often yield worse exact designs than heuristics that do not require approximate designs at all. In this paper, we build on an alternative principle of utilizing optimal approximate designs for the computation of optimal, or nearly-optimal, exact designs. The principle, which we call ascent with quadratic assistance (AQuA), is an integer programming method based on the quadratic approximation of the design criterion in the neighborhood of the optimal approximate information matrix. To this end, we present quadratic approximations of all Kiefer's criteria with an integer parameter, including D- and A-optimality and, by a model transformation, I-optimality. Importantly, we prove a low-rank property of the associated quadratic forms, which enables us to apply AQuA to large design spaces, for example via mixed integer conic quadratic solvers. We numerically demonstrate the robustness and superior performance of the proposed method for models under various types of constraints. More precisely, we compute optimal size-constrained exact designs for the model of spring-balance weighing, and optimal symmetric marginally restricted exact designs for the Scheffe mixture model. We also show how can iterative application of AQuA be used for a stratified information-based subsampling of large datasets under a lower bound on the quality and an upper bound on the cost of the subsample.