Duality-based Higher-order Non-smooth Optimization on Manifolds

We propose a 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. Numerical experiments on l2-TV-like problems confirm superlinear convergence on manifolds with positive and negative curvature