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A study of distributionally robust mixed-integer programming with Wasserstein metric: on the value of incomplete data

Sergey S. Ketkov

TL;DR

This work addresses MILPs with uncertain objective coefficients observed from incomplete data by formulating a three-level distributionally robust optimization problem. By centering a type-1 Wasserstein ball at the empirical data and constraining each sample with linear data-uncertainty, the authors derive an MILP reformulation under biaffine loss functions and adapt it to interval, semi-bandit, and bandit data contexts. Theoretical results show the three-level problem can be reduced to tractable single-level MILPs in key cases, while the computational study demonstrates how incomplete data can improve out-of-sample performance and how solution effort scales across problem classes. The approach provides a practical framework for data-driven MILPs when both data and its distribution are imperfectly known, with implications for online combinatorial optimization under uncertainty.

Abstract

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.

A study of distributionally robust mixed-integer programming with Wasserstein metric: on the value of incomplete data

TL;DR

This work addresses MILPs with uncertain objective coefficients observed from incomplete data by formulating a three-level distributionally robust optimization problem. By centering a type-1 Wasserstein ball at the empirical data and constraining each sample with linear data-uncertainty, the authors derive an MILP reformulation under biaffine loss functions and adapt it to interval, semi-bandit, and bandit data contexts. Theoretical results show the three-level problem can be reduced to tractable single-level MILPs in key cases, while the computational study demonstrates how incomplete data can improve out-of-sample performance and how solution effort scales across problem classes. The approach provides a practical framework for data-driven MILPs when both data and its distribution are imperfectly known, with implications for online combinatorial optimization under uncertainty.

Abstract

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.
Paper Structure (12 sections, 5 theorems, 71 equations, 5 figures, 1 algorithm)

This paper contains 12 sections, 5 theorems, 71 equations, 5 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related literature
  3. Our approach and contributions.
  4. Problem formulation
  5. Solution approach
  6. General case
  7. The case of interval uncertainty and semi-bandit feedback
  8. The case of bandit feedback
  9. Computational study
  10. Test instances and computational settings
  11. Results and discussion
  12. Conclusions

Key Result

Lemma 1

Assume that a decision $\mathbf{x} \in X$ and a data set $\widehat{\mathbf{C}}$ satisfying Assumption A3 are fixed. If, in addition, Assumption A2 holds, then the worst-case expectation problem admits an equivalent linear programming reformulation of the form:

Figures (5)

  • Figure 2: Let $K_{max} = 50$, $\gamma = \sqrt{K_{max}}$ and $h = 5$. We report the average relative loss (\ref{['eq: nominal relative loss']}) with MADs as a function of $K$ for $100$ random test instances.
  • Figure 3: The SPP with $h = 11$, $r = 5$ and $K_{max} = 100$. For different types of feedback we report the average relative loss (\ref{['eq: nominal relative loss']}) with MADs as a function of the sample size, $K$, for $100$ random test instances. In the case of bandit feedback the average solution times and the average LP relaxation quality with MADs are also provided.
  • Figure 5: The SPP with $r = 5$, $K = 50$ and $\varepsilon_K = \frac{\sqrt{2}}{11}h$. We report the average relative loss (\ref{['eq: nominal relative loss']}), the average solution times and the average LP relaxation quality with MADs as a function of $h$ for $100$ random test instances.
  • Figure 6: The MCP with $n_2 = 50$, $\tilde{h} = 5$, $K = 25$ and $\varepsilon_K = \frac{\sqrt{2}}{50}n_1$. We report the average relative loss (\ref{['eq: nominal relative loss']}), the average solution times and the average LP relaxation quality with MADs as a function of $n_1$ for $100$ random test instances.
  • Figure :

Theorems & Definitions (11)

  • Definition 1
  • Lemma 1
  • proof
  • Theorem 1
  • proof
  • Lemma 2
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • ...and 1 more