Data Privacy in Multi-Agent Optimization Under Uncertainty

We consider multi-agent optimization problems over networks, where agents cooperate to reach agreement on a common decision, or on the share of a common resource. This gives rise to the class of so called decision coupled and constraint coupled problems, respectively. We investigate the setting where the environment within which agents operate is uncertain, affecting agents' objective functions and/or constraint sets. We provide a data driven framework to address this problem in which, however, we view data as a finite and private resource. This implies that each agent has access to a finite amount of data (e.g., scenarios/realizations of the uncertain phenomenon), and these data neither are common nor are shared among agents, but constitute a private resource. Using tools from statistical learning and randomized optimization based on the so called scenario approach, we show how to accompany decisions in such a private-data regime with probabilistic certificates of their robustness when it comes into uncertainty realizations different from those included in the data. Moreover, we show that the class of decision coupled and constraint coupled problems naturally fits in this framework, and hence discuss general architectures and privacy preserving algorithms for computing decisions that enjoy such robustness certificates in a distributed manner.