Elfving's Theorem is a major result in the theory of optimal experimental design, which gives a geometrical characterization of c−optimality. In this paper, we extend this theorem to the case of multiresponse experiments, and we show that when the number of experiments is finite, the c−, A−, T − and D−optimal design of multiresponse experiments can be computed by Second-Order Cone Programming (SOCP). Moreover, the present SOCP approach can deal with design problems in which the variable is subject to several linear constraints.We give two proofs of this generalization of Elfving's theorem. One is based on Lagrangian dualization techniques and relies on the fact that the semidefinite programming (SDP) formulation of the multiresponse c−optimal design always has a solution which is a matrix of rank 1. Therefore, the complexity of this problem fades.We also investigate a model robust generalization of c−optimality, for which an Elfving-type theorem was established by Dette (1993). We show with the same Lagrangian approach that these model robust designs can be computed efficiently by minimizing a geometric mean under some norm constraints. Moreover, we show that the optimality conditions of this geometric programming problem yield an extension of Dette's theorem to the case of multiresponse experiments.When the goal is to identify a small number of linear functions of the unknown parameter (typically for c−optimality), we show by numerical examples that the present approach can be between 10 and 1000 times faster than the classic, state-of-the-art algorithms.

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.

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