An Adaptive Cubic Regularization quasi-Newton Method on Riemannian Manifolds

2024/02/19 by Maurício Silva Louzeiro, Louzeiro, Mauricio S., Gilson N. Silva +5 · 1 citation
Engineering · Mathematics · #Advanced Numerical Analysis Techniques #FOS: Mathematics #Iterative Methods for Nonlinear Equations #Numerical methods in inverse problems #Optimization and Control (math.OC)

paper · pdf · doi:10.48550/arxiv.2402.12464

openalex publication_date 2024/02/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

A quasi-Newton method with cubic regularization is designed for solving Riemannian unconstrained nonconvex optimization problems. The proposed algorithm is fully adaptive with at most \cal O (ε_g^-3/2) iterations to achieve a gradient smaller than ε_g for given ε_g, and at most \mathcal O(max\ ε_g^-(3)/(2), ε_H^-3 \) iterations to reach a second-order stationary point respectively. Notably, the proposed algorithm remains applicable even in cases of the gradient and Hessian of the objective function unknown. Numerical experiments are performed with gradient and Hessian being approximated by forward finite-differences to illustrate the theoretical results and numerical comparison.

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• A Cubic Regularization Method for Multiobjective Optimization

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