Work

A Primal-Dual Interior Point Trust Region Method for Second-Order Stationary Points of Riemannian Inequality-Constrained Optimization Problems

Year: 2026

Type: article

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.

Source: SIAM Journal on Optimization

DOI: 10.1137/25m1763871

Authors: Mitsuaki Obara, Takayuki Okuno, Akiko Takeda

Institutions: RIKEN, RIKEN Center for Advanced Intelligence Project, The University of Tokyo, Seikei University

Language: English

Topic: Advanced Optimization Algorithms Research

Subfield: Numerical Analysis

Field: Mathematics

Domain: Physical Sciences

Open Access status: green

Funder: Japan Society for the Promotion of Science

Awards: Development of Riemannian constrained optimization theory and applications, Development of nonlinear semidefinite optimization theory and application to machine learning, G5311064576, Development of large-scale bilevel optimization algorithms and their applications to machine learning