  In this paper, we present two novel manifold inexact augmented Lagrangian
methods, \textbf{ManIAL} for deterministic settings and \textbf{StoManIAL} for
stochastic settings, solving nonsmooth manifold optimization problems. By using
the Riemannian gradient method as a subroutine, we establish an
$\mathcal{O}(\epsilon^{-3})$ oracle complexity result of \textbf{ManIAL},
matching the best-known complexity result. Our algorithm relies on the careful
selection of penalty parameters and the precise control of termination criteria
for subproblems. Moreover, for cases where the smooth term follows an
expectation form, our proposed \textbf{StoManIAL} utilizes a Riemannian
recursive momentum method as a subroutine, and achieves an oracle complexity of
$\tilde{\mathcal{O}}(\epsilon^{-3.5})$, which surpasses the best-known
$\mathcal{O}(\epsilon^{-4})$ result. Numerical experiments conducted on sparse
principal component analysis and sparse canonical correlation analysis
demonstrate that our proposed methods outperform an existing method with the
previously best-known complexity result. To the best of our knowledge, these
are the first complexity results of the augmented Lagrangian methods for
solving nonsmooth manifold optimization problems.
