Probabilistic feasibility guarantees for solution sets to uncertain variational inequalities

We develop a data-driven approach to the computation of a-posteriori feasibility certificates for sets of solutions of variational inequalities affected by uncertainty. Specifically, we focus on variational inequalities with a deterministic mapping and an uncertain feasible 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 ones that typically impose that the solution set is a singleton and require certain non-degeneracy properties: hence, we thereby offer probabilistic feasibility guarantees for any feasible solution of the underlying variational inequality. We show that assessing the violation probability of an entire set of solutions requires enumeration of the support constraints that “shape” this set, and also propose a procedure to enumerate the support constraints that does not require a description of the solution set. We illustrate our results through numerical simulations on a robust game involving an electric vehicle charging coordination problem.