  We consider the minimization or maximization of the $J$th largest eigenvalue
of an analytic and Hermitian matrix-valued function, and build on Mengi et al.
(2014, SIAM J. Matrix Anal. Appl., 35, 699-724). This work addresses the
setting when the matrix-valued function involved is very large. We describe
subspace procedures that convert the original problem into a small-scale one by
means of orthogonal projections and restrictions to certain subspaces, and that
gradually expand these subspaces based on the optimal solutions of small-scale
problems. Global convergence and superlinear rate-of-convergence results with
respect to the dimensions of the subspaces are presented in the infinite
dimensional setting, where the matrix-valued function is replaced by a compact
operator depending on parameters. In practice, it suffices to solve eigenvalue
optimization problems involving matrices with sizes on the scale of tens,
instead of the original problem involving matrices with sizes on the scale of
thousands.
