We consider a general class of two-stage distributionally robust optimization (DRO) problems where the ambiguity set is constrained by fixed marginal probability laws that are not necessarily discrete. We derive primal and dual formulations of this class of problems and subsequently develop a numerical algorithm for computing approximate optimizers as well as approximate worst-case probability measures. Moreover, our algorithm computes both an upper bound and a lower bound for the optimal value of the problem, where the difference between the computed bounds provides a direct sub-optimality estimate of the computed solution. Most importantly, the sub-optimality can be controlled to be arbitrarily close to 0 by appropriately choosing the inputs of the algorithm. To demonstrate the effectiveness of the proposed algorithm, we apply it to three prominent instances of two-stage DRO problems in task scheduling, multi-product assembly, and supply chain network design with edge failure. The ambiguity sets in these problem instances involve a large number of continuous or discrete marginals. The numerical results showcase that the proposed algorithm computes high-quality robust decisions along with non-conservative sub-optimality estimates.

翻译：我们考虑一类广义的两阶段分布鲁棒优化问题，其模糊集受限于固定的边缘概率律，且这些边缘律不一定是离散的。我们推导了此类问题的原始形式与对偶形式，并随后开发了一种数值算法，用于计算近似最优解以及近似最坏情况概率测度。此外，该算法同时计算问题最优值的上界与下界，所计算边界之间的差值直接提供了所求解的次优性估计。最重要的是，通过适当选择算法的输入参数，次优性可以被控制到任意接近0。为验证所提算法的有效性，我们将其应用于任务调度、多产品装配以及考虑边失效的供应链网络设计这三个典型的两阶段DRO问题实例中。这些实例的模糊集涉及大量连续或离散的边缘分布。数值结果表明，所提算法能够计算出高质量的鲁棒决策，并给出非保守的次优性估计。