This paper focuses on solving mathematical programs with complementarity constraints (MPCCs) by assuming neither MPCC linear independence constraint qualification (MPCC-LICQ) nor lower/upper level strict complementarity at the solution. First, necessary conditions for MPCC local optimality and sufficient conditions for convergence to B-stationarity are investigated. Under MPCC-Abadie constraint qualification (MPCC-ACQ), we show that a local minimizer of an MPCC is “piecewise M-stationary”; a weakly stationary point of an MPCC is B-stationary if the related linear program with complementarity constraints (LPCC) is bounded below; furthermore, B-stationarity can be obtained from piecewise M-stationarity. Then convergence properties of the Bounding Algorithm proposed in [29] are analyzed. C- and M- stationarity of a limit point generated by the method are developed; an inequality variant of this method offers an alternative viewpoint to understand the behavior when approaching a limit point which is not S-stationary. In addition, a few practical issues related to convergence to a non-strongly stationary solution are discussed.