Probabilistic feasibility in data-driven multi-agent non-convex optimization

In this paper, we focus on the optimal operation of a multi-agent system affected by uncertainty. In particular, we consider a cooperative setting where agents jointly optimize a performance index compatibly with individual constraints on their discrete and continuous decision variables and with coupling global constraints. We assume that individual constraints are affected by uncertainty, which is known to each agent via a private set of data that cannot be shared with others. Exploiting tools from statistical learning theory, we provide data-based probabilistic feasibility guarantees for a (possibly sub-optimal) solution of the multi-agent problem that is obtained via a decentralized/distributed scheme that preserves the privacy of the local information. The generalization properties of the data-based solution are shown to depend on the size of each local dataset and on the complexity of the uncertain individual constraint sets. Explicit bounds are derived in the case of linear individual constraints. A comparative analysis with the cases of a common dataset and of local uncertainties that are independent is performed.