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

Mitsuaki Obara, Takayuki Okuno, Akiko Takeda

TL;DR

This work extends Riemannian optimization to nonlinear inequality-constrained problems by introducing RIPTRM, a primal-dual interior-point trust-region method operating on tangent spaces of the manifold. The method integrates a log-barrier merit with a trust-region subproblem solved via Riemannian Newton steps, and it establishes global convergence to AKKT points and SOSPs, with SOSP guarantees under second-order conditions. The authors implement two inner-solver strategies (a truncated CG and an eigenvalue-based exact solver) and provide numerical evidence showing higher accuracy and robustness relative to existing Riemannian interior-point methods, particularly in challenging Hessian spectra. The work also includes a comprehensive set of appendices delivering the full convergence proofs, sufficiency conditions, and additional experiments, underscoring the practical viability and theoretical rigor of RIPTRM for constrained optimization on manifolds.

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 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 find solutions with higher accuracy compared to an existing Riemannian interior point method and other algorithms. 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.

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

TL;DR

This work extends Riemannian optimization to nonlinear inequality-constrained problems by introducing RIPTRM, a primal-dual interior-point trust-region method operating on tangent spaces of the manifold. The method integrates a log-barrier merit with a trust-region subproblem solved via Riemannian Newton steps, and it establishes global convergence to AKKT points and SOSPs, with SOSP guarantees under second-order conditions. The authors implement two inner-solver strategies (a truncated CG and an eigenvalue-based exact solver) and provide numerical evidence showing higher accuracy and robustness relative to existing Riemannian interior-point methods, particularly in challenging Hessian spectra. The work also includes a comprehensive set of appendices delivering the full convergence proofs, sufficiency conditions, and additional experiments, underscoring the practical viability and theoretical rigor of RIPTRM for constrained optimization on manifolds.

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 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 find solutions with higher accuracy compared to an existing Riemannian interior point method and other algorithms. 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.
Paper Structure (33 sections, 35 theorems, 122 equations, 3 figures, 2 algorithms)

This paper contains 33 sections, 35 theorems, 122 equations, 3 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Our contribution
  3. Paper organization
  4. Preliminaries
  5. Notation and terminology from Riemannian geometry
  6. Optimality conditions for RICO
  7. Proposed method
  8. Overview of RIPTRM
  9. Outer iteration
  10. Inner iteration
  11. Global convergence analysis
  12. Global convergence of outer iterations
  13. Consistency of inner iterations
  14. Global convergence of inner iteration
  15. Numerical experiments
  16. ...and 18 more sections

Key Result

Proposition 2.1

\newlabelprop:secodrretrHess0 If the retraction $\mathop{\mathrm{R}}\nolimits_{}$ is second order, then, for any $\theta^{}\in\mathfrak{F}\lparen*\rparen{\mathcal{M}}$, where the left-hand side is the Hessian of $\hat{\theta^{}}_{x_{}}\colon T_{x_{}}\mathcal{M}\to\mathbb{R}^{}$ at $0_{x_{}}\in T_{x_{}}\mathcal{M}$.

Figures (3)

  • Figure 1: Box plots of best residuals among 20 instances
  • Figure 1: Residual over time for Rosenbrock function minimization
  • Figure 2: Residuals over time on a representative case

Theorems & Definitions (75)

  • Proposition 2.1: Absiletal08OptimAlgoonMatMani
  • Definition 2.2: YangZhangSong14OptimCondforNLPonRiemMani
  • Theorem 2.3: YangZhangSong14OptimCondforNLPonRiemMani
  • Theorem 2.4: YamakawaSato2022SeqOptimCondforNLOonRiemManiandGlobConvALM
  • Proposition 2.5: YamakawaSato2022SeqOptimCondforNLOonRiemManiandGlobConvALM
  • Definition 2.6: ObaraOkunoTakeda2021SQOforNLOonRiemMani
  • Theorem 2.7: YangZhangSong14OptimCondforNLPonRiemMani
  • Remark 2.8
  • Definition 4.1
  • Theorem 4.2
  • ...and 65 more