Data-driven Stochastic Programming with Distributionally Robust Constraints Under Wasserstein Distance: Asymptotic Properties

Distributionally robust optimization is a dominant paradigm for decision-making problems where the distribution of random variables is unknown. We investigate a distributionally robust optimization problem with ambiguities in the objective function and countably infinite constraints. The ambiguity set is defined as a Wasserstein ball centered at the empirical distribution. Based on the concentration inequality of Wasserstein distance, we establish the asymptotic convergence property of the data-driven distributionally robust optimization problem when the sample size goes to infinity. We show that with probability 1, the optimal value and the optimal solution set of the data-driven distributionally robust problem converge to those of the stochastic optimization problem with true distribution. Finally, we provide numerical evidences for the established theoretical results.