Spectral Residual Method for Nonlinear Equations on Riemannian Manifolds

TL;DR

This paper introduces a spectral residual method adapted for solving nonlinear equations on Riemannian manifolds, enhancing convergence speed and efficiency through adaptive spectral parameters and a non-monotone globalization technique.

Contribution

The paper develops a novel spectral residual algorithm for Riemannian manifolds, incorporating adaptive spectral parameters and proving its global convergence.

Findings

Effective in solving tangent vector fields on various Riemannian manifolds.

Outperforms existing methods like the Polak-Ribière-Polyak method.

Demonstrates high efficiency and convergence speed.

Abstract

In this paper, the spectral algorithm for nonlinear equations (SANE) is adapted to the problem of finding a zero of a given tangent vector field on a Riemannian manifold. The generalized version of SANE uses, in a systematic way, the tangent vector field as a search direction and a continuous real-valued function that adapts this direction and ensures that it verifies a descent condition for an associated merit function. In order to speed up the convergence of the proposed method, we incorporate a Riemannian adaptive spectral parameter in combination with a non-monotone globalization technique. The global convergence of the proposed procedure is established under some standard assumptions. Numerical results indicate that our algorithm is very effective and efficient solving tangent vector field on different Riemannian manifolds and competes favorably with a Polak-Ribi\'ere-Polyak Method recently published and other methods existing in the literature.