Cooperative data-driven distributionally robust optimization

This paper studies a class of multiagent stochastic optimization problems where the objective is to minimize the expected value of a function which depends on a random variable. The probability distribution of the random variable is unknown to the agents, so each one gathers samples of it. The agents aim to cooperatively find, using their data, a solution to the optimization problem with guaranteed out-of-sample performance. The approach is to formulate a data-driven distributionally robust optimization problem using Wasserstein ambiguity sets, which turns out to be equivalent to a convex program. We reformulate the latter as a distributed optimization problem and identify a convex-concave augmented Lagrangian function whose saddle points are in correspondence with the optimizers provided a min-max interchangeability criteria is met. Our distributed algorithm design then consists of the saddle-point dynamics associated to the augmented Lagrangian. We formally establish that the trajectories of the dynamics converge asymptotically to a saddle point and hence an optimizer of the problem. Finally, we provide a class of functions that meet the min-max interchangeability criteria. Simulations illustrate our results.