  We present an algorithm based on continuation techniques that can be applied
to solve numerically minimization problems with equality constraints. We focus
on problems with a great number of local minima which are hard to obtain by
local minimization algorithms with random starting guesses. We are particularly
interested in the computation of minimal norm solutions of underdetermined
systems of polynomial equations. Such systems arise, for instance, in the
context of the construction of high order optimized differential equation
solvers. By applying our algorithm, we are able to obtain 10th order
time-symmetric composition integrators with smaller 1-norm than any other
integrator found in the literature up to now.
