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

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, $c-,A-,T-$ and $D-$optimal design of multiresponse experiments can be computed by Second-Order Cone Programming (SOCP). Moreover, our 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 \emph{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 number of unknown parameters is small, or when the number of linear functions of the parameters to be estimated is small, we show by numerical examples that our approach can be between 10 and 1000 times faster than the classic, state-of-the-art algorithms.