  Optimization problems over compact Lie groups have been extensively studied
due to their broad applications in linear programming and optimal control. This
paper analyzes least square problems over a noncompact Lie group, the
symplectic group $\Sp(2N,\R)$, which can be used to assess the optimality of
control over dynamical transformations in classical mechanics and quantum
optics. The critical topology for minimizing the Frobenius distance from a
target symplectic transformation is solved. It is shown that the critical
points include a unique local minimum and a number of saddle points. The
topology is more complicated than those of previously studied problems on
compact Lie groups such as the orthogonal and unitary groups because the
incompatibility of the Frobenius norm with the pseudo-Riemannian structure on
the symplectic group brings significant nonlinearity to the problem.
Nonetheless, the lack of traps guarantees the global convergence of local
optimization algorithms.
