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Let the design of an experiment be represented by an $s$-dimensional vector$\mathbf {w}$ of weights with nonnegative components. Let the quality of$\mathbf {w}$ for the estimation of the parameters of the statistical model bemeasured by the criterion of $D$-optimality, defined as the $m$th root of thedeterminant of the information matrix $M(\mathbf {w})=\sum_{i=1}^sw_iA_iA_i^T$,where $A_i,i=1,\ldots,s$ are known matrices with $m$ rows. In this paper, weshow that the criterion of $D$-optimality is second-order cone representable.As a result, the method of second-order cone programming can be used to computean approximate $D$-optimal design with any system of linear constraints on thevector of weights. More importantly, the proposed characterization allows us tocompute an exact $D$-optimal design, which is possible thanks to high-qualitybranch-and-cut solvers specialized to solve mixed integer second-order coneprogramming problems. Our results extend to the case of the criterion of$D_K$-optimality, which measures the quality of $\mathbf {w}$ for theestimation of a linear parameter subsystem defined by a full-rank coefficientmatrix $K$. We prove that some other widely used criteria are also second-ordercone representable, for instance, the criteria of $A$-, $A_K$-, $G$- and$I$-optimality. We present several numerical examples demonstrating theefficiency and general applicability of the proposed method. We show that inmany cases the mixed integer second-order cone programming approach allows usto find a provably optimal exact design, while the standard heuristicssystematically miss the optimum.

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