This study addresses a class of linear mixed-integer programming (MILP) problems that involve uncertainty in the objective function parameters. The parameters are assumed to form a random vector, whose 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. First, we prove that the three-level problem admits a single-level MILP reformulation, if the class of loss functions is restricted to biaffine functions. Secondly, it turns out that for several particular forms of data uncertainty, the outlined problem can be solved reasonably fast by leveraging the nominal MILP problem. Finally, we conduct a computational study, where the out-of-sample performance of our model and computational complexity of the proposed MILP reformulation are explored numerically for several application domains.

翻译：本研究针对目标函数参数存在不确定性的一类线性混合整数规划（MILP）问题。假设参数构成随机向量，其概率分布仅能通过有限训练数据集观测。与现有文献中多数相关研究不同，我们同时考虑了基础数据集的不确定性。数据不确定性通过每个随机样本的线性约束集合描述，而（针对固定数据实现）分布的不确定性则基于以数据经验分布为中心的1-Wasserstein球定义。整体问题被构建为三层分布鲁棒优化（DRO）问题。首先，我们证明当损失函数类别限制为双仿射函数时，该三层问题可转化为单层MILP重构。其次，针对数据不确定性的若干特殊形式，可通过利用标称MILP问题实现合理快速求解。最后，我们开展计算研究，在多个应用领域中对所提模型的样本外性能及MILP重构的计算复杂度进行数值探索。