A Dai-Yuan-type Riemannian conjugate gradient method with the weak Wolfe conditions

TL;DR

This paper introduces a new Riemannian conjugate gradient method based on Dai-Yuan's Euclidean algorithm, which guarantees global convergence under weak Wolfe conditions, supported by theoretical analysis and numerical experiments.

Contribution

It generalizes Dai-Yuan's Euclidean conjugate gradient method to Riemannian manifolds, ensuring global convergence with weak Wolfe conditions, unlike previous methods requiring strong Wolfe conditions.

Findings

The proposed method is globally convergent under weak Wolfe conditions.

Numerical experiments confirm the effectiveness of the new algorithm.

The method extends Dai-Yuan's approach to Riemannian optimization.

Abstract

This article describes a new Riemannian conjugate gradient method and presents a global convergence analysis. The existing Fletcher-Reeves-type Riemannian conjugate gradient method is guaranteed to be globally convergent if it is implemented with the strong Wolfe conditions. On the other hand, the Dai-Yuan-type Euclidean conjugate gradient method generates globally convergent sequences under the weak Wolfe conditions. This article deals with a generalization of Dai-Yuan's Euclidean algorithm to a Riemannian algorithm that requires only the weak Wolfe conditions. The global convergence property of the proposed method is proved by means of the scaled vector transport associated with the differentiated retraction. The results of numerical experiments demonstrate the effectiveness of the proposed algorithm.