Consistency analysis of a local Lipschitz homeomorphism of an SAA normal mapping for a parametric stochastic variational inequality

We investigate the consistency of a local Lipschitz homeomorphism of the normal mapping of a sample average approximation (SAA) parametric stochastic variational inequality. Since Thibault directional derivatives are unbounded, the notion of cosmic deviation is introduced to characterize the total outer limit of a sequence of Thibault directional derivatives of SAA mappings. We demonstrate that when the normal mapping of the original problem is locally Lipschitz homeomorphic, the normal mapping of the SAA problem is sufficient if the sample size is large enough. We use this result to develop the consistency of stationary points of the SAA estimator for a stochastic mathematical program with complementarity constraints.