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In the past years, augmented Lagrangian methods have been successfullyapplied to several classes of non-convex optimization problems, inspiring newdevelopments in both theory and practice. In this paper we bring most of theserecent developments from nonlinear programming to the context of optimizationon Riemannian manifolds, including equality and inequality constraints. Manyresearch have been conducted on optimization problems on manifolds, howeveronly recently the treatment of the constrained case has been considered. Inthis paper we propose to bridge this gap with respect to the most recentdevelopments in nonlinear programming. In particular, we formulate several wellknown constraint qualifications from the Euclidean context which are sufficientfor guaranteeing global convergence of augmented Lagrangian methods, withoutrequiring boundedness of the set of Lagrange multipliers. Convergence of thedual sequence can also be assured under a weak constraint qualification. Thetheory presented is based on so called sequential optimality conditions, whichis a powerful tool used in this context. The paper can also be read with theEuclidean context in mind, serving as a review of the most relevant constraintqualifications and global convergence theory of state-of-the-art augmentedLagrangian methods for nonlinear programming.

Constraint qualifications and strong global convergence properties of an augmented Lagrangian method on Riemannian manifolds