This study addresses a class of linear mixed-integer programming (MIP) problems that involve uncertainty in the objective function coefficients. The coefficients are assumed to form a random vector, which probability distribution can only be observed through a finite training data set. Unlike most of the related studies in the literature, we also consider uncertainty in the underlying data set. The data uncertainty is described by a set of linear constraints for each random sample, and the uncertainty in the distribution (for a fixed realization of data) is defined using a type-1 Wasserstein ball centered at the empirical distribution of the data. The overall problem is formulated as a three-level distributionally robust optimization (DRO) problem. We prove that for a class of bi-affine loss functions the three-level problem admits a linear MIP reformulation. Furthermore, it turns out that in several important particular cases the three-level problem can be solved reasonably fast by leveraging the nominal MIP problem. Finally, we conduct a computational study, where the out-of-sample performance of our model and computational complexity of the proposed MIP reformulation are explored numerically for several application domains.

翻译：本研究针对一类目标函数系数存在不确定性的线性混合整数规划（MIP）问题。系数被假设构成随机向量，其概率分布仅能通过有限训练数据集观测。与文献中大多数相关研究不同，我们同时考虑底层数据集的不确定性。数据不确定性通过每个随机样本的一组线性约束描述，而分布不确定性（针对数据的固定实现）则采用以数据经验分布为中心的1型Wasserstein球定义。整体问题被表述为三层分布鲁棒优化（DRO）问题。我们证明对于一类双仿射损失函数，该三层问题可转化为线性MIP重表述。进一步研究表明，在若干重要特例中，通过利用名义MIP问题可较快求解该三层问题。最后，我们开展计算研究，数值探索了所提模型在多个应用领域的样本外性能及MIP重表述的计算复杂度。