Abadie-type constraint qualification for mathematical programs with equilibrium constraints.

*(English)*Zbl 1090.90200Summary: Mathematical programs with equilibrium constraints (MPEC) are nonlinear programs which do not satisfy any of the common constraint qualifications (CQ). In order to obtain first-order optimality conditions, constraint qualifications tailored to the MPECs have been developed and researched in the past. In this paper, we introduce a new Abadie-type constraint qualification for MPECs. We investigate sufficient conditions for this new CQ, discuss its relationship to several existing MPEC constraint qualifications, and introduce a new Slater-type constraint qualifications. Finally, we prove a new stationarity concept to be a necessary optimality condition under our new Abadie-type CQ.

##### MSC:

90C46 | Optimality conditions and duality in mathematical programming |

90C33 | Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming) |

##### Keywords:

Mathematical programs with equilibrium constraints; Abadie constraint qualification; Slater constraint qualification; optimality conditions
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\textit{M. L. Flegel} and \textit{C. Kanzow}, J. Optim. Theory Appl. 124, No. 3, 595--614 (2005; Zbl 1090.90200)

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##### References:

[5] | Flegel, M. L., and Kanzow, C., On the Guignard Constraint Qualification for Mathematical Programs with Equilibrium Constraints, Preprint 248, Institute of Applied Mathematics and Statistics, University of Würzburg, 2002. · Zbl 1147.90397 |

[11] | Mangasarian, O. L., Nonlinear Programming, SIAM, Philadelphia, Pennsylvania, 1994. |

[14] | Flegel, M. L., and Kanzow, C., An Abadie- Type Constraint Qualification for Mathematical Programs with Equilibrium Constraints, Preprint, Institute of Applied Mathematics and Statistics, University of Würzburg, 2002. · Zbl 1090.90200 |

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