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

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## Abstract

Depending on whether a mathematical program with equilibrium constraints (MPEC) is considered in its original or its enhanced (via KKT conditions) form, the assumed qualification conditions as well as the derived necessary optimality conditions may differ significantly. In this paper, we study this issue when imposing one of the weakest possible qualification conditions, namely the calmness of the perturbation mapping associated with the respective generalized equations in both forms of the MPEC. It is well known that the calmness property allows one to derive the so-called M-stationarity conditions. The restrictiveness of assumptions and the strength of conclusions in the two forms of the MPEC is also strongly related to the qualification conditions on the “lower level”. For instance, even under the linear independence constraint qualification (LICQ) for a lower level feasible set described by \(\mathscr {C}^1\) functions, the calmness properties of the original and the enhanced perturbation mapping are drastically different. When passing to \(\mathscr {C}^{1,1}\) data, this difference still remains true under the weaker Mangasarian–Fromovitz constraint qualification, whereas under LICQ both the calmness assumption and the derived optimality conditions are fully equivalent for the original and the enhanced form of the MPEC. After clarifying these relations, we provide a compilation of practically relevant consequences of our analysis in the derivation of necessary optimality conditions. The obtained results are finally applied to MPECs with structured equilibria.

## Keywords

Mathematical programs with equilibrium constraints Optimality conditions Constraint qualification Calmness Perturbation mapping## Mathematics Subject Classification

65K10 90C30 90C31 90C46## Notes

### Acknowledgements

The authors would like to thank two anonymous referees for their critical comments which led to a substantial improvement of the paper. We are particularly indebted to Prof. Helmut Gfrerer for pointing us to an incorrect result in the first version of the paper.

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