  We consider optimization problems on manifolds with equality and inequality
constraints. A large body of work treats constrained optimization in Euclidean
spaces. In this work, we consider extensions of existing algorithms from the
Euclidean case to the Riemannian case. Thus, the variable lives on a known
smooth manifold and is further constrained. In doing so, we exploit the growing
literature on unconstrained Riemannian optimization. For the special case where
the manifold is itself described by equality constraints, one could in
principle treat the whole problem as a constrained problem in a Euclidean
space. The main hypothesis we test here is whether it is sometimes better to
exploit the geometry of the constraints, even if only for a subset of them.
Specifically, this paper extends an augmented Lagrangian method and smoothed
versions of an exact penalty method to the Riemannian case, together with some
fundamental convergence results. Numerical experiments indicate some gains in
computational efficiency and accuracy in some regimes for minimum balanced cut,
non-negative PCA and $k$-means, especially in high dimensions.
