ABSTRACT. In the paper we solve the problem of D H -optimal design on a discrete experimental domain, which is formally equivalent to maximizing determinant on the convex hull of a finite set of positive semidefinite matrices. The problem of D H -optimality covers many special design settings, e.g., the D-optimal experimental design for multivariate regression models. For D H -optimal designs we prove several theorems generalizing known properties of standard D-optimality. Moreover, we show that D H -optimal designs can be numerically computed using a multiplicative algorithm, for which we give a proof of convergence. We illustrate the results on the problem of D-optimal augmentation of independent regression trials for the quadratic model on a rectangular grid of points in the plane.
In the paper, we solve the problem of computing the maximin efficient design with respect to the class of all orthogonally invariant criteria. It turns out that on a finite experimental domain, the maximin efficient design can be computed by the methods of semidefinite programming. Using this approach, we can deal with the non-differentiability inherent in the problem, due to which the standard iterative procedures cannot be applied. We illustrate the results on the models of polynomial regression on a line segment and quadratic regression on a cube.
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