We improve the inequality used in (Pronzato, 2003) to remove points from the design space during the search for a D-optimum design. Let ξ be any design on a compact space X ⊂ R m with a nonsingular information matrix, and let m + ǫ be the maximum of the variance function d(ξ, x) over all x ∈ X. We prove that any support point x * of a D-optimum design on X must satisfy the inequality d(ξ, x *) ≥ m(1 + ǫ/2 − ǫ(4 + ǫ − 4/m)/2). We show that this new lower bound on d(ξ, x *) is, in a sense, the best possible, and how it can be used to accelerate algorithms for D-optimum design.

Let the design of an experiment be represented by an sdimensional vector w of weights with nonnegative components. Let the quality of w for the estimation of the parameters of the statistical model be measured by the criterion of D-optimality, defined as the mth root of the determinant of the information matrix M (w) = s 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 matrix K.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.

We propose a class of subspace ascent methods for computing optimal approximate designs that covers both existing as well as new and more efficient algorithms. Within this class of methods, we construct a simple, randomized exchange algorithm (REX). Numerical comparisons suggest that the performance of REX is comparable or superior to the performance of state-ofthe-art methods across a broad range of problem structures and sizes. We focus on the most commonly used criterion of D-optimality that also has applications beyond experimental design, such as the construction of the minimum volume ellipsoid containing a given set of data-points. For D-optimality, we prove that the proposed algorithm converges to the optimum. We also provide formulas for the optimal exchange of weights in the case of the criterion of A-optimality. These formulas enable one to use REX for computing A-optimal and I-optimal designs.

No abstract