  We extend the classical primal-dual interior point method from the Euclidean
setting to the Riemannian one. Our method, named the Riemannian interior point
method, is for solving Riemannian constrained optimization problems. We
establish its local superlinear and quadratic convergence under the standard
assumptions. Moreover, we show its global convergence when it is combined with
a classical line search. Our method is a generalization of the classical
framework of primal-dual interior point methods for nonlinear nonconvex
programming. Numerical experiments show the stability and efficiency of our
method.
