In this paper we consider two problems of frame theory. On the one hand,
given a fixed frame ${\mathcal F}$ we describe explicitly the spectral and
geometric structure of optimal frames ${\mathcal W}$ that are in duality with
${\mathcal F}$ and such that the Frobenius norms of their analysis operators is
bounded from below by a fixed constant, where optimality is measured with
respect to submajorization. On the other hand, given a set of vectors
${\mathcal F}$ we describe the spectral and geometrical structure of optimal
completions of ${\mathcal F}$ by a finite family of vectors with prescribed
norms, under certain hypothesis. Again, optimality is measured with respect to
majorization of the frames operators, which implies optimality with respect to
a family of convex functionals that include the mean square error and the
Bendetto-Fickus' potential. Our approach relies on the description of the
spectral and geometrical structure of matrices that minimize submajorization on
sets that are naturally associated with the problems above.