Scenario Reduction for Distributionally Robust Optimization

Abstract: Stochastic and (distributionally) robust optimization problems often become computationally challenging as the number of scenarios increases. Scenario reduction is therefore a key technique for improving tractability. We introduce a general scenario reduction method for distributionally robust optimization (DRO), which includes stochastic and robust optimization as special cases. Our approach constructs the reduced DRO problem by projecting the original ambiguity set onto a reduced set of scenarios. Under mild conditions, we establish bounds on the relative quality of the reduction. The methodology is applicable to random variables following either discrete or continuous probability distributions, with representative scenarios appropriately selected in both cases. Given the relevance of optimization problems with linear and quadratic objectives, we further refine our approach for these settings. Finally, we demonstrate its effectiveness through numerical experiments on mixed-integer benchmark instances from MIPLIB and portfolio optimization problems. Our results show that the proposed approximation significantly reduces solution time while maintaining high solution quality with only minor errors.