Modified memoryless spectral-scaling Broyden family on Riemannian manifolds

TL;DR

This paper introduces a modified memoryless quasi-Newton method on Riemannian manifolds that enhances search directions and demonstrates global convergence, showing promising numerical performance for specific optimization problems.

Contribution

It proposes a novel modification to the spectral-scaling Broyden family on Riemannian manifolds, including a new parameter and a general map, with proven convergence and numerical validation.

Findings

The method satisfies the sufficient descent condition.

It achieves global convergence under Wolfe conditions.

Numerical results favor the BFGS-based approach for off-diagonal minimization.

Abstract

This paper presents modified memoryless quasi-Newton methods based on the spectral-scaling Broyden family on Riemannian manifolds. The method involves adding one parameter to the search direction of the memoryless self-scaling Broyden family on the manifold. Moreover, it uses a general map instead of vector transport. This idea has already been proposed within a general framework of Riemannian conjugate gradient methods where one can use vector transport, scaled vector transport, or an inverse retraction. We show that the search direction satisfies the sufficient descent condition under some assumptions on the parameters. In addition, we show global convergence of the proposed method under the Wolfe conditions. We numerically compare it with existing methods, including Riemannian conjugate gradient methods and the memoryless spectral-scaling Broyden family. The numerical results indicate that the proposed method with the BFGS formula is suitable for solving an off-diagonal cost function minimization problem on an oblique manifold.