This paper focuses on the critical points of the positive semidefinite (PSD) rank regularized minimization problem and its equivalent reformulations, including the mathematical program formulation with equilibrium constraint (MPEC), the global exact penalty of the MPEC, the DC surrogate problem yielded by eliminating the dual part. We disclose the relationship among these critical points so as to lend a full leverage to the user for choosing an appropriate reformulation to seek a low-rank solution.As a byproduct, we provide a weaker condition for a local minimizer to be the M-stationary point of the MPEC by characterizing the directional limiting normal cone to the graph of the normal cone mapping ofthe PSD cone.
