  We address the problem of computing the smallest symplectic eigenvalues and
the corresponding eigenvectors of symmetric positive-definite matrices in the
sense of Williamson's theorem. It is formulated as minimizing a trace cost
function over the symplectic Stiefel manifold. We first investigate various
theoretical aspects of this optimization problem such as characterizing the
sets of critical points, saddle points, and global minimizers as well as
proving that non-global local minimizers do not exist. Based on our recent
results on constructing Riemannian structures on the symplectic Stiefel
manifold and the associated optimization algorithms, we then propose solving
the symplectic eigenvalue problem in the framework of Riemannian optimization.
Moreover, a connection of the sought solution with the eigenvalues of a special
class of Hamiltonian matrices is discussed. Numerical examples are presented.
