  Non-smoothness at optimal points is a common phenomenon in many eigenvalue
optimization problems. We consider two recent algorithms to minimize the
largest eigenvalue of a Hermitian matrix dependent on one parameter, both
proven to be globally convergent unaffected by non-smoothness. One of these
models the eigenvalue function with a piece-wise quadratic function, and
effective in dealing with non-convex problems. The other projects the Hermitian
matrix into subspaces formed of eigenvectors, and effective in dealing with
large-scale problems. We generalize the latter slightly to cope with
non-smoothness. For both algorithms, we analyze the rate-of-convergence in the
non-smooth setting, when the largest eigenvalue is multiple at the minimizer
and zero is strictly in the interior of the generalized Clarke derivative, and
prove that both algorithms converge rapidly. The algorithms are applied to, and
the deduced results are illustrated on the computation of the inner numerical
radius, the modulus of the point on the boundary of the field of values closest
to the origin, which carries significance for instance for the numerical
solution of a definite generalized symmetric eigenvalue problem.
