Stochastic mathematical programs with equilibrium constraints.

*(English)*Zbl 0937.90076Summary: We introduce stochastic mathematical programs with equilibrium constraints (SMPEC), which generalize MPEC models by explicitly incorporating possible uncertainties in the problem data to obtain robust solutions to hierarchical problems. For this problem, we establish results on the existence of solutions, and on the convexity and directional differentiability of the implicit upper-level objective function, both for continuously and discretely distributed probability distributions. In so doing, we establish links between SMPEC models and two-stage stochastic programs with recourse. We also discuss basic parallel iterative algorithms for discretely distributed SMPEC problems.

##### MSC:

90C15 | Stochastic programming |

47J20 | Variational and other types of inequalities involving nonlinear operators (general) |

49J40 | Variational inequalities |

##### Keywords:

bilevel programming; variational inequality problems; stochastic programming; existence of solutions
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\textit{M. Patriksson} and \textit{L. Wynter}, Oper. Res. Lett. 25, No. 4, 159--167 (1999; Zbl 0937.90076)

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