Data-driven distributed optimization using Wasserstein ambiguity sets

This paper considers a general class of stochastic optimization problem for multiagent systems. We assume that the probability distribution of the uncertain parameters is unknown to the agents and instead, each agent gathers a certain number of samples of it. The objective for the agents is to cooperatively find, using the available data, a solution that has performance guarantees for the stochastic problem. To this end, we formulate a data-driven distributionally robust optimization (DRO) problem using Wasserstein ambiguity sets that has the desired performance guarantees. With the aim of solving this optimization in a distributed manner, we identify a convex-concave modified Lagrangian function whose saddle points are in correspondence with the optimizers of the DRO problem. We then design our distributed algorithm as the gradient descent in the convex variable and gradient ascent in the concave variable of this Lagrangian function. Our convergence analysis shows that the trajectories of this dynamics converge asymptotically to an optimizer of the DRO problem. Simulations illustrate our results.