A primal-dual interior point trust region method for second-order stationary points of Riemannian inequality-constrained optimization problems

TL;DR

This paper introduces RIPTRM, a novel Riemannian primal-dual interior point trust region method that guarantees convergence to second-order stationary points for inequality-constrained problems on manifolds.

Contribution

It is the first algorithm combining trust region strategies with Riemannian constraints, achieving second-order convergence for nonlinear inequality-constrained optimization.

Findings

RIPTRM converges globally to approximate KKT points and weak second-order stationary points.

Numerical experiments demonstrate high accuracy and promising performance of RIPTRM.

Exact search directions improve performance in cases with large negative Hessian eigenvalues.

Abstract

We consider Riemannian inequality-constrained optimization problems. Such problems inherit the benefits of Riemannian approach developed in the unconstrained setting and naturally arise from applications in control, machine learning, and other fields. We propose a Riemannian primal-dual interior point trust region method (RIPTRM) for solving them. We prove its global convergence to an approximate Karush-Kuhn-Tucker point and a weak second-order stationary point. Under the strict complementarity condition, this result reduces to global convergence to a second-order stationary point. To the best of our knowledge, this is the first algorithm that incorporates the trust region strategy for constrained optimization on Riemannian manifolds, and has the second-order convergence property for optimization problems on Riemannian manifolds with nonlinear inequality constraints. We conduct numerical experiments in which we introduce a truncated conjugate gradient method and an eigenvalue-based subsolver for RIPTRM to approximately and exactly solve the trust region subproblems, respectively. Empirical results show that RIPTRMs consistently find solutions with high accuracy. Additionally, we observe that RIPTRM with the exact search direction shows promising performance in an instance where the Hessian of the Lagrangian has a large negative eigenvalue.