Conjugate Gradient Methods for Optimization Problems on Symplectic Stiefel Manifold

Abstract

The symplectic Stiefel manifold is a Riemannian manifold that is a generalization of the symplectic group. In this study, we propose novel conjugate gradient methods on the symplectic Stiefel manifold and compare them with the steepest descent method proposed in existing studies through numerical experiments. Although the theoretical basis of the Riemannian conjugate gradient methods has already been established, special treatment is required to address specific manifolds since these methods utilize some mappings, such as a retraction and vector transport, on the manifold. Numerical experiments demonstrate that the proposed method outperforms existing methods and is efficient.