Optimization on Riemannian manifolds, also called Riemannian optimization, considers finding an optimum of a real-valued function defined on a Riemannian manifold. Riemannian optimization has been a topic of much interest over the past few years due to many important applications. In this presentation, the framework of Riemannian optimization is introduced, and the current state of Riemannian optimization algorithms are briefly reviewed. To show a research focus and difficulties of Riemannian optimization, we generalize the proximal Newton method to the Riemannian setting and the difficulties therein are highlighted. The convergence results are given. Numerical experiments verify that the Riemannian proximal Newton method converges superlinearly locally.
