A collection of efficient retractions for the symplectic Stiefel manifold

This article introduces a new map on the symplectic Stiefel manifold. The operation that requires the highest computational cost to compute the novel retraction is a inversion of size 2p-by-2p, which is much less expensive than those required for the available retractions in the literature. Later, with the new retraction, we design a constraint preserving gradient method to minimize smooth functions defined on the symplectic Stiefel manifold. To improve the numerical performance of our approach, we use the non-monotone line-search of Zhang and Hager with an adaptive Barzilai–Borwein type step-size. Our numerical studies show that the proposed procedure is computationally promising and is a very good alternative to solve large-scale optimization problems over the symplectic Stiefel manifold.