  This work concerns the global minimization of a prescribed eigenvalue or a
weighted sum of prescribed eigenvalues of a Hermitian matrix-valued function
depending on its parameters analytically in a box. We describe how the
analytical properties of eigenvalue functions can be put into use to derive
piece-wise quadratic functions that underestimate the eigenvalue functions.
These piece-wise quadratic under-estimators lead us to a global minimization
algorithm, originally due to Breiman and Cutler. We prove the global
convergence of the algorithm, and show that it can be effectively used for the
minimization of extreme eigenvalues, e.g., the largest eigenvalue or the sum of
the largest specified number of eigenvalues. This is particularly facilitated
by the analytical formulas for the first derivatives of eigenvalues, as well as
analytical lower bounds on the second derivatives that can be deduced for
extreme eigenvalue functions. The applications that we have in mind also
include the ${\rm H}_\infty$-norm of a linear dynamical system, numerical
radius, distance to uncontrollability and various other non-convex eigenvalue
optimization problems, for which, generically, the eigenvalue function involved
is simple at all points.
