An Inexact Semismooth Newton Method on Riemannian Manifolds with Application to Duality-Based Total Variation Denoising

We propose a higher-order method for solving nonsmooth optimization problems on manifolds. To obtain superlinear convergence, we apply a Riemannian semismooth Newton method to a nonsmooth nonlinear 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 semismooth Newton method and prove conditions for local linear and superlinear convergence that hold independent of the sign of the curvature. Numerical experiments on l(2)-TV-like problems with dual regularization confirm superlinear convergence on manifolds with positive and negative curvature.