A class of spectral conjugate gradient methods for Riemannian optimization

Abstract

Spectral conjugate gradient (SCG) methods are combinations of spectral gradient method and conjugate gradient (CG) methods, which have been well studied in Euclidean space. In this paper, we aim to extend this class of methods to solve optimization problems on Riemannian manifolds. Firstly, we present a Riemannian version of the spectral parameter, which guarantees that the search direction always satisfies the sufficient descent property without the help of any line search strategy. Secondly, we introduce a generic algorithmic framework for the Riemannian SCG methods, in which the selection of the CG parameter is very flexible. Under the Riemannian Wolfe conditions, the global convergence of the proposed algorithmic framework is established whenever the absolute value of the CG parameter is no more than the Riemannian Fletcher–Reeves CG parameter. Finally, some preliminary numerical results are reported and compared with several classical Riemannian CG methods, which show that our new methods are efficient.