  In this study we analyze linear mixed-integer programming problems, in which
the distribution of the cost vector is only observable through a finite
training data set. In contrast to the related studies, we assume that the
number of random observations for each component of the cost vector may vary.
Then the goal is to find a prediction rule that converts the data set into an
estimate of the expected value of the objective function and a prescription
rule that provides an associated estimate of the optimal decision. We aim at
finding the least conservative prediction and prescription rules, which satisfy
some specified asymptotic guarantees as the sample size tends to infinity. We
demonstrate that under some mild assumption the resulting vector optimization
problems admit a Pareto optimal solution with some attractive theoretical
properties. In particular, this solution can be obtained by solving a
distributionally robust optimization (DRO) problem with respect to all
probability distributions with given component-wise relative entropy distances
from the empirical marginal distributions. It turns out that the outlined DRO
problem can be solved rather effectively whenever there exists an effective
algorithm for the respective deterministic problem. In addition, we perform
numerical experiments where the out-of-sample performance of the proposed
approach is analyzed.
