Computing exact D-optimal designs by mixed integer second-order cone programming

i=1 wiAiA T i , where Ai,i = 1,...,s are known matrices with m rows. In this paper, we show that the criterion of D-optimality is secondorder cone representable. As a result, the method of second-order cone programming can be used to compute an approximate D-optimal design with any system of linear constraints on the vector of weights. More importantly, the proposed characterization allows us to compute an exact D-optimal design, which is possible thanks to highquality branch-and-cut solvers specialized to solve mixed integer second-order cone programming problems. Our results extend to the case of the criterion of DK-optimality, which measures the quality of w for the estimation of a linear parameter subsystem defined by a full-rank coefficient matrixK. We prove that some other widely used criteria are also secondorder cone representable, for instance, the criteria of A-, AK-, Gand I-optimality. We present several numerical examples demonstrating the efficiency and general applicability of the proposed method. We show that in many cases the mixed integer second-order cone programming approach allows us to find a provably optimal exact design, while the standard heuristics systematically miss the optimum.

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