  For the rank regularized minimization problem, we introduce several kinds of
stationary points by the problem itself and its equivalent reformulations
including the mathematical program with an equilibrium constraint (MPEC), the
global exact penalty of the MPEC,the surrogate yielded by eliminating the dual
part in the exact penalty. A clear relation chart is established for these
stationary points, which guides the user to choose an appropriate reformulation
for seeking a low-rank solution. As a byproduct, we also provide a weaker
condition for a local minimizer of the MPEC to be the M-stationary point by
characterizing the directional limiting normal cone to the graph of the normal
cone mapping of the positive semidefinite (PSD) cone.
