Proximal Point Method for Quasiconvex Functions in Riemannian Manifolds

Abstract

This paper studies the convergence of the proximal point method for quasiconvex functions in finite dimensional complete Riemannian manifolds. We prove initially that, in the general case, when the objective function is proper and lower semicontinuous, each accumulation point of the sequence generated by the method, if it exists, is a limiting critical point of the function. Then, under the assumptions that the sectional curvature of the manifold is bounded above by some non negative constant and the objective function is quasiconvex we analyze two cases. When the constant is zero, the global convergence of the algorithm to a limiting critical point is assured and if it is positive, we prove the local convergence for a class of quasiconvex functions, which includes Lipschitz functions.