Local near-quadratic convergence of Riemannian interior point methods

TL;DR

This paper analyzes Riemannian interior point methods, demonstrating local superlinear and near-quadratic convergence under standard assumptions, and introduces a specific barrier update for improved convergence rates.

Contribution

It provides the first Riemannian constrained optimization algorithm achieving both local superlinear and near-quadratic convergence with second-order stationarity analysis.

Findings

The algorithm achieves local superlinear convergence by solving a linear system at each outer iteration.

A specific barrier parameter update yields near-quadratic convergence.

Numerical results support the theoretical convergence analyses.

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 parameter that achieves local near-quadratic convergence of the algorithm. We apply our results to the method proposed by Obara, Okuno, and Takeda (2026) 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. Numerical results support the theoretical analyses of the proposed methods.