  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.
