A scenario-based approach to multi-agent optimization with distributed information

In this paper, we consider optimization problems involving multiple agents. Each agent introduces its own constraints on the optimization vector, and the constraints of all agents depend on a common source of uncertainty. We suppose that uncertainty is known locally to each agent through a private set of data (multi-agent scenarios), and that each agent enforces its scenario-based constraints to the solution of the multi-agent optimization problem. Our goal is to assess the feasibility properties of the corresponding multi-agent scenario solution. In particular, we are able to provide a priori certificates that the solution is feasible for a new occurrence of the global uncertainty with a probability that depends on the size of the datasets and the desired confidence level. The recently introduced wait-and-judge approach to scenario optimization and the notion of support rank are used for this purpose. Notably, decision-coupled and constraint-coupled uncertain optimization programs for multi-agent systems fit our framework and, hence, any distributed optimization scheme to solve the associated multi-agent scenario problem can be accompanied with our a priori probabilistic feasibility certificates.