An accelerated spectral CG based algorithm for optimization techniques on Riemannian manifolds and its comparative evaluation

Abstract

The conjugate gradient (CG) method and its accelerated variants are an efficient class of methods for solving unconstrained optimization problems in Euclidean space. This paper aims to develop an accelerated spectral CG (SCG) method for solving optimization problems on Riemannian manifolds. A general algorithmic framework for the accelerated Riemannian SCG (ARSCG) method is presented, in which a general transport mapping is introduced and the Riemannian spectral parameter ensures that the search direction is always a descent direction of the objective function. By adjusting the stepsize of the Riemannian CG method, we enhance the rate of descent of the objective function. The global convergence of the algorithm is established under the assumption that the absolute value of the CG parameter does not exceed the Fletcher–Reeves CG parameter. Moreover, a linear convergence rate is demonstrated under the assumption that the objective function is geodesically strongly convex. Finally, some preliminary numerical results are reported, indicating the proposed algorithm ARSCG performs well numerically compared to some related Riemannian CG methods.