Wasserstein Distributionally Robust Nash Equilibrium Seeking with Heterogeneous Data: A Lagrangian Approach

We study a class of distributionally robust games where agents are allowed to heterogeneously choose their risk aversion with respect to distributional shifts of the uncertainty. In our formulation, heterogeneous Wasserstein ball constraints on each distribution are enforced through a penalty function leveraging a Lagrangian formulation. We then formulate the distributionally robust Nash equilibrium problem and show that under certain assumptions it is equivalent to a finite-dimensional variational inequality problem with a strongly monotone mapping. We then design an approximate Nash equilibrium seeking algorithm and prove convergence of the average regret to a quantity that diminishes with the number of iterations, thus learning the desired equilibrium up to an a priori specified accuracy. Numerical simulations corroborate our theoretical findings.

Adversarially and Distributionally Robust Virtual Energy Storage Systems via the Scenario Approach

We propose an optimization model where a parking lot manager (PLM) can aggregate parked EV batteries to provide virtual energy storage services that are provably robust under uncertain EV departures and state-of-charge caps. Our formulation yields a data-driven convex optimization problem where a prosumer community agrees on a contract with the PLM for the provision of storage services over a finite horizon. Leveraging recent results in the scenario approach, we certify out-of-sample constraint safety. Furthermore, we enable a tunable profit-risk trade-off through scenario relaxation and extend our model to account for robustness to adversarial perturbations and distributional shifts over Wasserstein-based ambiguity sets. All the approaches are accompanied by tight finite-sample certificates. Numerical studies demonstrate the out-of-sample and out-of-distribution constraint satisfaction of our proposed model compared to the developed theoretical guarantees, showing their effectiveness and potential in robust and efficient virtual energy services.