  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 (RIPM), is for solving Riemannian constrained optimization problems. We establish its locally superlinear and quadratic convergence under the standard assumptions. Moreover, we show its global convergence when it is combined with a classical line search. This method is a generalization of the classical framework of primal-dual interior point methods for nonlinear programming proposed by El-Bakry et al. in 1996. Numerical experiments show the stability and efficiency of our method. 
Riemannian Interior Point Methods for Constrained Optimization on   Manifolds