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

TL;DR

This paper extends augmented Lagrangian methods to Riemannian manifolds with constraints, establishing conditions for global convergence and dual sequence convergence, bridging a gap between nonlinear programming and manifold optimization.

Contribution

It introduces constraint qualifications ensuring global convergence of augmented Lagrangian methods on Riemannian manifolds, including cases without bounded Lagrange multipliers.

Findings

Constraint qualifications guarantee global convergence.

Dual sequence convergence under weak constraint qualifications.

Framework applicable to both Euclidean and Riemannian optimization.

Abstract

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.