  We deal with the minimization of the ${\mathcal H}_\infty$-norm of the
transfer function of a parameter-dependent descriptor system over the set of
admissible parameter values. Subspace frameworks are proposed for such
minimization problems where the involved systems are of large order. The
proposed algorithms are greedy interpolatory approaches inspired by our recent
work [Aliyev et al., SIAM J. Matrix Anal. Appl., 38(4):1496--1516, 2017] for
the computation of the ${\mathcal H}_\infty$-norm. In this work, we minimize
the ${\mathcal H}_\infty$-norm of a reduced-order parameter-dependent system
obtained by two-sided restrictions onto certain subspaces. Then we expand the
subspaces so that Hermite interpolation properties hold between the full and
reduced-order system at the optimal parameter value for the reduced order
system. We formally establish the superlinear convergence of the subspace
frameworks under some smoothness assumptions. The fast convergence of the
proposed frameworks in practice is illustrated by several large-scale systems.
