The Proximal Point Algorithm in Riemannian Manifolds

O. P. Ferreira and Oliveira, P. R.

In this paper we consider the minimization problem with constraints. We will show that if the set of constraints is a Riemannian manifold of nonpositive sectional curvature, and the objective function is convex in this manifold, then the proximal point method in Euclidean space is naturally extended to solve minimization problems in Riemannian manifolds. We will prove that the sequence generated by our method is well defined and converges to a minimizer point. In particular we show how tools of Riemannian geometry, more specifically the convex analysis in Riemannian manifolds, can be used to solve nonconvex constrained problems in Euclidean spaces.

T. R. ES-453 PESC/COPPE Federal University of Rio de Janeiro 09/97, revised 10/99