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：本研究针对一类目标函数参数包含不确定性的线性混合整数规划（MILP）问题。假设参数构成一个随机向量，其概率分布仅能通过有限训练数据集观测。与文献中多数相关研究不同，我们同时考虑了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线性约束集合描述，而分布不确定性（针对数据固定实现）则采用以数据经验分布为中心的1型Wasserstein球来定义。整体问题被建模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重构形式。其次，研究发现对于若干特定形式的数据不确定性，通过利用名义MILP问题可相当快速地求解所述问题。最后开展计算研究，通过多个应用领域数值实验，从样本外性能与所提MILP重构的计算复杂度两个维度对我们提出的模型进行探索。