Local near-quadratic convergence of Riemannian interior point methods
Mitsuaki Obara, Takayuki Okuno, Akiko Takeda
TL;DR
This work develops and analyzes a Riemannian interior-point framework for RNLO on a manifold, introducing an extrapolation-based initialization that enables outer iterations to converge with no inner iterations in a neighborhood of a solution. It proves local superlinear convergence of the outer loop and, with a carefully chosen barrier-parameter update, local near-quadratic convergence, under LICQ and SOSC. The results are specialized to a RIPTRM for RNLO with $\mathcal{E}=\emptyset$, establishing both local SOSP convergence and compatibility with global SOSP convergence. Overall, this constitutes a first approach to achieving both local and global convergence to SOSPs for constrained optimization on Riemannian manifolds, with practical implications for manifold-constrained problems in diverse applications.
Abstract
We consider Riemannian optimization problems with inequality and equality constraints and analyze a class of Riemannian interior point methods for solving them. The algorithm of interest consists of outer and inner iterations. We show that, under standard assumptions, the algorithm achieves local superlinear convergence by solving a linear system at each outer iteration, removing the need for further computations in the inner iterations. We also provide a specific update for the barrier parameters that achieves local near-quadratic convergence of the algorithm. We apply our results to the method proposed by Obara, Okuno, and Takeda (2025) and show its local superlinear and near-quadratic convergence with an analysis of the second-order stationarity. To our knowledge, this is the first algorithm for constrained optimization on Riemannian manifolds that achieves both local convergence and global convergence to a second-order stationary point.