This article introduces a new retraction on the symplectic Stiefel manifold. The operation that requires the highest computational cost to compute the novel retraction is a matrix 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. In order 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.
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