On M-stationarity conditions in MPECs and the associated qualification conditions

Depending on whether a mathematical program with equilibrium constraints 6 (MPEC) is considered in its original or its enhanced (via KKT conditions) form, the assumed 7 constraint qualifications (CQs) as well as the derived necessary optimality conditions may 8 differ significantly. In this paper, we study this issue when imposing one of the weakest pos9 sible CQs, namely the calmness of the perturbation mapping associated with the respective 10 generalized equations in both forms of the MPEC. It is well known that the calmness prop11 erty allows one to derive so-called M-stationarity conditions. The strength of assumptions 12 and conclusions in the two forms of the MPEC is strongly related with the CQs on the ’lower 13 level’ imposed on the set whose normal cone appears in the generalized equation. For in14 stance, under just the Mangasarian-Fromovitz CQ (a minimum assumption required for this 15 set), the calmness properties of the original and the enhanced perturbation mapping are dras16 tically different. They become identical in the case of a polyhedral set or when adding the 17 Full Rank CQ. On the other hand, the resulting optimality conditions are affected too. If the 18 considered set even satisfies the Linear Independence CQ, both the calmness assumption 19 and the derived optimality conditions are fully equivalent for the original and the enhanced 20 form of the MPEC. A compilation of practically relevant consequences of our analysis in 21 the derivation of necessary optimality conditions is provided in the main Theorem 7. The 22 obtained results are finally applied to MPECs with structured equilibria. 23