grobner basis solution for optimization?

One of my favorite semi-definite programming problems is to minimize
the maximal eigenvalue of a linear combination

F0 - x1 * F1 - x2 * F2 - ... - xn * Fn,

where the Fi are square symmetric matrices with integer entries. It is
clear that the answer is an algebraic number, and I can often recover
it using PSLQ following a high-precision computation of the minimum.

But it would be better to use Grobner bases. Does anyone know of any
work like that--it would seem that the system of polynomial equations
to be solved is quite horrid.

--Ron Bruck
.