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 oroposed approximation significantly reduces solution time while maintaining high solution quality with only minor errors.
We propose a novel online learning framework for robust Bayesian optimization of uncertain black-box functions. While Bayesian optimization is well-suited for data-efficient optimization of expensive objectives, its standard form can be sensitive to hidden or varying parameters. To address this issue, we consider a min–max robust counterpart of the optimization problem and develop a practically efficient solution algorithm, BROVER (Bayesian Robust Optimization via Exploration with Regret minimization). Our method combines Gaussian process regression with a decomposition approach: the minimax structure is split into a non-convex online learner based on the Follow-the-Perturbed-Leader algorithm together with a subsequent minimization step in the decision variables. We prove that the theoretical regret bound converges under mild assumptions, ensuring asymptotic convergence to robust solutions. Numerical experiments on synthetic data validate the regret guarantees and demonstrate fast convergence to the robust optimum. Furthermore, we apply our method to the robust optimization of organic solar cell performance, where hidden process parameters and experimental variability naturally induce uncertainty. Our results on real-world datae show that BROVER identifies solutions with strong robustness properties within relatively few iterations, thereby offering a modern and practical approach for data-driven black-box optimization under uncertainty.