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

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