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, c- and A-optimal designs of multiresponse experiments can be computed by Second-Order Cone Programming (SOCP). Our new approach allows us to solve much larger instances, by comparison with the semidefinite programming (SDP) based approaches which are considered to be the state of the art on this problem. We give two proofs of this result. One is based on Lagrangian dualization techniques and shows that the SDP formulation of the multiresponse c-optimal design always has a solution which is a matrix of rank 1, which explains why the complexity of this problem fades. Then, we investigate a "model robust" generalization of c-optimal designs, 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 geometrical programming problem yield an extension of Dette's theorem to the case of multiresponse experiments. We illustrate our results with numerical experiments.
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