  In the past years, augmented Lagrangian methods have been successfully
applied to several classes of non-convex optimization problems, inspiring new
developments in both theory and practice. In this paper we bring most of these
recent developments from nonlinear programming to the context of optimization
on Riemannian manifolds, including equality and inequality constraints. Many
research have been conducted on optimization problems on manifolds, however
only recently the treatment of the constrained case has been considered. In
this paper we propose to bridge this gap with respect to the most recent
developments in nonlinear programming. In particular, we formulate several well
known constraint qualifications from the Euclidean context which are sufficient
for guaranteeing global convergence of augmented Lagrangian methods, without
requiring boundedness of the set of Lagrange multipliers. Convergence of the
dual sequence can also be assured under a weak constraint qualification. The
theory presented is based on so called sequential optimality conditions, which
is a powerful tool used in this context. The paper can also be read with the
Euclidean context in mind, serving as a review of the most relevant constraint
qualifications and global convergence theory of state-of-the-art augmented
Lagrangian methods for nonlinear programming.
