  Optimization of convex functions subject to eigenvalue constraints is
intriguing because of peculiar analytical properties of eigenvalues, and is of
practical interest because of wide range of applications in fields such as
structural design and control theory. Here we focus on the optimization of a
linear objective subject to a constraint on the smallest eigenvalue of an
analytical and Hermitian matrix-valued function. We offer a quadratic support
function based numerical solution. The quadratic support functions are derived
utilizing the variational properties of an eigenvalue over a set of Hermitian
matrices. Then we establish the local convergence of the algorithm under mild
assumptions, and deduce a precise rate of convergence result by viewing the
algorithm as a fixed point iteration. We illustrate its applicability in
practice on the pseudospectral functions.
