Riemannian Interior Point Methods for Constrained Optimization on Manifolds

Abstract

We extend the classical primal-dual interior point algorithms from Euclidean setting to Riemannian setting. This is named as Riemannian Interior Point (RIP) Method for the Riemannian constrained optimization problem. Under the standard assumptions in Riemannian setting, we establish the locally superlinear, quadratic convergence for Newton version of RIP, and the locally linear, superlinear convergence for quasi-Newton version of RIP. Those are the generalizations of classical local convergence theory of primal-dual interior point algorithms for nonlinear programming, proposed by El-Bakry et al. in 1996, andYamashita et al. in 1996.