Data-Driven Combinatorial Optimization with Incomplete Information: a Distributionally Robust Optimization Approach

In this study we analyze linear combinatorial optimization problems where the cost vector is not known a priori, but is only observable through a finite data set. In contrast to the related studies, we presume that the number of observations with respect to particular components of the cost vector may vary. The goal is to find a procedure that transforms the data set into an estimate of the expected value of the objective function (which is referred to as a prediction rule) and a procedure that retrieves a candidate decision (which is referred to as a prescription rule). We aim at finding the least conservative prediction and prescription rules, which satisfy some specified asymptotic guarantees. We demonstrate that the resulting vector optimization problems admit a weakly optimal solution, which can be obtained by solving a particular distributionally robust optimization problem. Specifically, the decision-maker may optimize the worst-case expected loss across all probability distributions with given component-wise relative entropy distances from the empirical marginal distributions. Finally, we perform numerical experiments to analyze the out-of-sample performance of the proposed solution approach.