Ascent with Quadratic Assistance for the Construction of Exact Experimental Designs
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In the area of experimental design, there is a large body of theoretical knowledge and computational experience concerning so-called optimal approximate designs. However, for an approximate design to be executed in a practical setting, it must be converted into an exact design, which is usually done via rounding procedures. Although generally rapid, rounding procedures have several drawbacks; in particular, they often yield worse exact designs than heuristics that do not require approximate designs at all. In this paper, we propose an alternative method of utilizing optimal approximate designs for the computation of optimal, or nearly-optimal, exact designs. The proposed method, which we call ascent with quadratic assistance (AQuA), is a hill-climbing method in which the information matrix of an optimal approximate design is used to formulate a quadratic approximation of the design criterion. To this end, we present quadratic approximations of Kiefer's criteria with an integer parameter, and we prove the low-rank property of the associated quadratic forms. We numerically demonstrate the generally superior performance of AQuA relative to both rounding procedures and standard heuristics for a problem of optimal mixture experimental design as well as for randomly generated regression models.
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