The symplectic Stiefel manifold, denoted by ${Sp}(2p,2n)$, is the set of linear maps between standard spaces $\mathbb{R}^{2p}$ and $\mathbb{R}^{2n}$. When $p=n$, it reduces to well-known $2n\times 2n$ matrices. Optimization problems on ${Sp}(2p,2n)$ find applications in various areas, such as optics, quantum physics, numerical algebra, model order reduction dynamical systems. purpose this paper...