Probabilistic feasibility guarantees for solution sets to uncertain variational inequalities

TL;DR

This paper introduces a data-driven method to provide probabilistic feasibility guarantees for the entire solution set of uncertain variational inequalities, extending previous approaches that focused on single solutions.

Contribution

It extends scenario-based robustness analysis to entire solution sets of variational inequalities with uncertainty, without requiring non-degeneracy assumptions or closed-form solutions.

Findings

Provides probabilistic guarantees for solution sets under uncertainty

Introduces a support constraint enumeration procedure for complex sets

Demonstrates applicability through electric vehicle charging case study

Abstract

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.