Optimal Transport-Based Distributionally Robust Optimization: Structural Properties and Iterative Schemes

We consider optimal transport-based distributionally robust optimization (DRO) problems with locally strongly convex transport cost functions and affine decision rules. Under conventional convexity assumptions on the underlying loss function, we obtain structural results about value policy, worst-case adversarial model. These expose a rich structure embedded in DRO problem (e.g., strong even if non-DRO is not convex, suitable scaling of Lagrangian for constraint, etc., which are crucial design efficient algorithms). As consequence these results, one can develop procedures that have same sample iteration complexity as natural benchmark algorithm, such stochastic gradient descent.