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**Optimality conditions for disjunctive programs based on generalized differentiation with application to mathematical programs with equilibrium constraints.**
*(English)*
Zbl 1298.49021

Summary: We consider optimization problems with a disjunctive structure of the constraints. Prominent examples of such problems are mathematical programs with equilibrium constraints or vanishing constraints. Based on the concepts of directional subregularity and their characterization by means of objects from generalized differentiation, we obtain the new stationarity concept of extended M-stationarity, which turns out to be an equivalent dual characterization of B-stationarity. These results are valid under a very weak constraint qualification of Guignard-type which is usually very difficult to verify. We also state a new constraint qualification which is a little bit stronger but verifiable. Further, we present second-order optimality conditions, both necessary and sufficient. Finally, we apply these results to the special case of mathematical programs with equilibrium constraints and compute explicitly all the objects from generalized differentiation. For this type of problem we also introduce the concept of strong M-stationarity which builds a bridge between S-stationarity and M-stationarity.

### MSC:

49J52 | Nonsmooth analysis |

49J53 | Set-valued and variational analysis |

49K21 | Optimality conditions for problems involving relations other than differential equations |

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

90C56 | Derivative-free methods and methods using generalized derivatives |