  Given a finite sequence of vectors $\mathcal F_0$ in $\C^d$ we characterize
in a complete and explicit way the optimal completions of $\mathcal F_0$
obtained by adding a finite sequence of vectors with prescribed norms, where
optimality is measured with respect to majorization (of the eigenvalues of the
frame operators of the completed sequence). Indeed, we construct (in terms of a
fast algorithm) a vector - that depends on the eigenvalues of the frame
operator of the initial sequence $\cF_0$ and the sequence of prescribed norms -
that is a minimum for majorization among all eigenvalues of frame operators of
completions with prescribed norms. Then, using the eigenspaces of the frame
operator of the initial sequence $\cF_0$ we describe the frame operators of all
optimal completions for majorization. Hence, the concrete optimal completions
with prescribed norms can be obtained using recent algorithmic constructions
related with the Schur-Horn theorem.
  The well known relation between majorization and tracial inequalities with
respect to convex functions allow to describe our results in the following
equivalent way: given a finite sequence of vectors $\mathcal F_0$ in $\C^d$ we
show that the completions with prescribed norms that minimize the convex
potential induced by a strictly convex function are structural minimizers, in
the sense that they do not depend on the particular choice of the convex
potential.
