  We develop a data-driven approach to the computation of a-posteriori
feasibility certificates to the solution sets of variational inequalities
affected by uncertainty. Specifically, we focus on instances of variational
inequalities with a deterministic mapping and an uncertain feasibility set, and
represent uncertainty by means of scenarios. Building upon recent advances in
the scenario approach literature, we quantify the robustness properties of the
entire set of solutions of a variational inequality, with feasibility set
constructed using the scenario approach, against a new unseen realization of
the uncertainty. Our results extend existing results that typically impose an
assumption that the solution set is a singleton and require certain
non-degeneracy properties, and thereby offer probabilistic feasibility
guarantees to any feasible solution. We show that assessing the violation
probability of an entire set of solutions, rather than of a singleton, requires
enumeration of the support constraints that "shape" this set. Additionally, we
propose a general procedure to enumerate the support constraints that does not
require a closed form description of the solution set, which is unlikely to be
available. We show that robust game theory problems can be modelling via
uncertain variational inequalities, and illustrate our theoretical results
through extensive numerical simulations on a case study involving an electric
vehicle charging coordination problem.
