Agent independent probabilistic robustness certificates for robust optimization programs with uncertain quadratic cost

This paper focuses on a specific class of convex multi-agent programs, prevalent in many practical applications, where agents cooperate to minimize a common cost, expressed as a function of the aggregate decision and affected by uncertainty. We model uncertainty by means of scenarios and use an epigraphic reformulation to transfer the uncertain part of the cost function to the constraints. Then, by exploiting the structure of the program under study and leveraging on existing results in the scenario approach literature, and in particular using the so called support rank notion, we provide for the optimal solution of the program distribution-free robustness certificates that are agent independent. This means that the constructed bound on the probability of constraint violation does not depend on the number of agents, but only on the dimension of the agents' decision. This leads to a significant improvement as it substantially reduces the number of samples required to achieve a certain level of probabilistic robustness as the number of agents increases. Our certificates can be used alongside any convex optimization algorithm centralised, decentralised or distributed, to obtain an optimal solution of the underlying problem. Our theoretical results are accompanied by a numerical example that investigates the electric vehicle charging problem and validates that the obtained robustness certificate is independent of the number of vehicles in the fleet.