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Elfving's Theorem is a major result in the theory of optimal experimentaldesign, which gives a geometrical characterization of $c-$optimality. In thispaper, we extend this theorem to the case of multiresponse experiments, and weshow that when the number of experiments is finite, $c-,A-,T-$ and $D-$optimaldesign of multiresponse experiments can be computed by Second-Order ConeProgramming (SOCP). Moreover, our SOCP approach can deal with design problemsin which the variable is subject to several linear constraints.  We give two proofs of this generalization of Elfving's theorem. One is basedon Lagrangian dualization techniques and relies on the fact that thesemidefinite programming (SDP) formulation of the multiresponse $c-$optimaldesign always has a solution which is a matrix of rank $1$. Therefore, thecomplexity of this problem fades.  We also investigate a \emph{model robust} generalization of $c-$optimality,for which an Elfving-type theorem was established by Dette (1993). We show withthe same Lagrangian approach that these model robust designs can be computedefficiently by minimizing a geometric mean under some norm constraints.Moreover, we show that the optimality conditions of this geometric programmingproblem yield an extension of Dette's theorem to the case of multiresponseexperiments.  When the number of unknown parameters is small, or when the number of linearfunctions of the parameters to be estimated is small, we show by numericalexamples that our approach can be between 10 and 1000 times faster than theclassic, state-of-the-art algorithms.

Computing Optimal Designs of multiresponse Experiments reduces to Second-Order Cone Programming