M-stationarity of Local Minimizers of MPCCs and Convergence of NCP-based Methods

This paper focuses on solving mathematical programs with complementarity constraints (MPCCs) without assuming MPCC-LICQ or lower level strict complementarity at a solution. We show that a local minimizer of an MPCC is “piecewise M-stationary” un- der MPCC-GCQ; furthermore, every weakly stationary point of an MPCC is B-stationary if MPCC-ACQ holds. For the Bounding Algorithm proposed in [22], which solves MPCCs via an NCP-based reformulation, we develop C- and M- stationarity of a limit point of the method by assuming only MPCC-GCQ. In particular, an inequality variant of this method offers an alternative viewpoint to understand the behavior of an algorithm when approach- ing a local minimizer of an MPCC which is not S-stationary. In addition, a few practical issues related to convergence to a non-strongly stationary solution are discussed, including a comparison between the behaviors of the NCP-based methods and of a typical regularization method, i.e., the REG method proposed in [19].