  We propose a higher-order method for solving non-smooth optimization problems
on manifolds. In order to obtain superlinear convergence, we apply a Riemannian
Semi-smooth Newton method to a non-smooth non-linear primal-dual optimality
system based on a recent extension of Fenchel duality theory to Riemannian
manifolds. We also propose an inexact version of the Riemannian Semi-smooth
Newton method and prove conditions for local linear and superlinear convergence
that hold independent of the sign of the curvature. Numerical experiments on
l2-TV-like problems with dual regularization confirm superlinear convergence on
manifolds with positive and negative curvature.
