New stationarity conditions between strong and M-stationarity for mathematical programs with complementarity constraints

We introduce new first-order necessary conditions for mathematical programs with complementarity constraints (MPCCs), which lie between strong and M-stationarity and have a relatively simple description. We show that they hold for local minimizers under the rather weak constraint qualification MPCC-GCQ. As a generalization, we also get a class of stationarity conditions that lie between strong and C-stationarity and show that they also hold for local minimizers under MPCC-GCQ. We also present similar results for mathematical programs with vanishing constraints (MPVCs), and a very simple and elementary proof of M-stationarity for local minimizers of MPVCs.