  Riemannian optimization is concerned with problems, where the independent
variable lies on a smooth manifold. There is a number of problems from
numerical linear algebra that fall into this category, where the manifold is
usually specified by special matrix structures, such as orthogonality or
definiteness. Following this line of research, we investigate tools for
Riemannian optimization on the symplectic Stiefel manifold. We complement the
existing set of numerical optimization algorithms with a Riemannian trust
region method tailored to the symplectic Stiefel manifold. To this end, we
derive a matrix formula for the Riemannian Hessian under a right-invariant
metric. Moreover, we propose a novel retraction for approximating the
Riemannian geodesics. Finally, we conduct a comparative study in which we
juxtapose the performance of the Riemannian variants of the steepest descent,
conjugate gradients, and trust region methods on selected matrix optimization
problems that feature symplectic constraints.
