  The symplectic Stiefel manifold, denoted by $\mathrm{Sp}(2p,2n)$, is the set
of linear symplectic maps between the standard symplectic spaces
$\mathbb{R}^{2p}$ and $\mathbb{R}^{2n}$. When $p=n$, it reduces to the
well-known set of $2n\times 2n$ symplectic matrices. Optimization problems on
$\mathrm{Sp}(2p,2n)$ find applications in various areas, such as optics,
quantum physics, numerical linear algebra and model order reduction of
dynamical systems. The purpose of this paper is to propose and analyze
gradient-descent methods on $\mathrm{Sp}(2p,2n)$, where the notion of gradient
stems from a Riemannian metric. We consider a novel Riemannian metric on
$\mathrm{Sp}(2p,2n)$ akin to the canonical metric of the (standard) Stiefel
manifold. In order to perform a feasible step along the antigradient, we
develop two types of search strategies: one is based on quasi-geodesic curves,
and the other one on the symplectic Cayley transform. The resulting
optimization algorithms are proved to converge globally to critical points of
the objective function. Numerical experiments illustrate the efficiency of the
proposed methods.
