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Numerical method for approximately optimal solutions of two-stage distributionally robust optimization with marginal constraints

Ariel Neufeld, Qikun Xiang

TL;DR

This work develops a numerically tractable framework for two-stage distributionally robust optimization with marginal-constrained ambiguity sets. By replacing the inner cost with its dual and leveraging a primal–dual saddle-point structure, the authors derive strong duality, existence of optimizers, and sub-optimality bounds. They then propose an iterative algorithm that solves a linear-semi-infinite-program (LSIP) via cutting planes and recovers approximate primal solutions and worst-case measures, with provable bounds that can be made arbitrarily tight. The method is demonstrated on three application domains—task scheduling, assemble-to-order, and supply chain network design with edge failure—showing high-quality robust decisions, non-conservative sub-optimality estimates, and informative structure in the worst-case distributions. Overall, the approach combines theoretical guarantees with a practical algorithmic toolkit (including a MIP-based global max step) for handling non-discrete marginals in large-scale DRO problems.

Abstract

We consider a general class of two-stage distributionally robust optimization (DRO) problems where the ambiguity set is constrained by fixed marginal probability laws that are not necessarily discrete. We derive primal and dual formulations of this class of problems and subsequently develop a numerical algorithm for computing approximate optimizers as well as approximate worst-case probability measures. Moreover, our algorithm computes both an upper bound and a lower bound for the optimal value of the problem, where the difference between the computed bounds provides a direct sub-optimality estimate of the computed solution. Most importantly, the sub-optimality can be controlled to be arbitrarily close to 0 by appropriately choosing the inputs of the algorithm. To demonstrate the effectiveness of the proposed algorithm, we apply it to three prominent instances of two-stage DRO problems in task scheduling, multi-product assembly, and supply chain network design with edge failure. The ambiguity sets in these problem instances involve a large number of continuous or discrete marginals. The numerical results showcase that the proposed algorithm computes high-quality robust decisions along with non-conservative sub-optimality estimates.

Numerical method for approximately optimal solutions of two-stage distributionally robust optimization with marginal constraints

TL;DR

This work develops a numerically tractable framework for two-stage distributionally robust optimization with marginal-constrained ambiguity sets. By replacing the inner cost with its dual and leveraging a primal–dual saddle-point structure, the authors derive strong duality, existence of optimizers, and sub-optimality bounds. They then propose an iterative algorithm that solves a linear-semi-infinite-program (LSIP) via cutting planes and recovers approximate primal solutions and worst-case measures, with provable bounds that can be made arbitrarily tight. The method is demonstrated on three application domains—task scheduling, assemble-to-order, and supply chain network design with edge failure—showing high-quality robust decisions, non-conservative sub-optimality estimates, and informative structure in the worst-case distributions. Overall, the approach combines theoretical guarantees with a practical algorithmic toolkit (including a MIP-based global max step) for handling non-discrete marginals in large-scale DRO problems.

Abstract

We consider a general class of two-stage distributionally robust optimization (DRO) problems where the ambiguity set is constrained by fixed marginal probability laws that are not necessarily discrete. We derive primal and dual formulations of this class of problems and subsequently develop a numerical algorithm for computing approximate optimizers as well as approximate worst-case probability measures. Moreover, our algorithm computes both an upper bound and a lower bound for the optimal value of the problem, where the difference between the computed bounds provides a direct sub-optimality estimate of the computed solution. Most importantly, the sub-optimality can be controlled to be arbitrarily close to 0 by appropriately choosing the inputs of the algorithm. To demonstrate the effectiveness of the proposed algorithm, we apply it to three prominent instances of two-stage DRO problems in task scheduling, multi-product assembly, and supply chain network design with edge failure. The ambiguity sets in these problem instances involve a large number of continuous or discrete marginals. The numerical results showcase that the proposed algorithm computes high-quality robust decisions along with non-conservative sub-optimality estimates.
Paper Structure (16 sections, 14 theorems, 78 equations, 8 figures, 2 algorithms)

This paper contains 16 sections, 14 theorems, 78 equations, 8 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Model for two-stage DRO problems with marginal constraints
  3. Settings
  4. Examples
  5. Primal and dual formulations
  6. Numerical algorithm
  7. Numerical experiments
  8. Experiment 1: task scheduling
  9. Experiment 2: multi-product assembly
  10. Experiment 3: supply chain network design with edge failure
  11. Discussions
  12. Proof of theoretical results
  13. Results about one-dimensional semi-discrete optimal transport
  14. Proof of results in Section \ref{['sec:dro']}
  15. Proof of results in Section \ref{['sec:primaldual']}
  16. ...and 1 more sections

Key Result

Lemma 2.2

Under Setting sett:dro, the following statements hold.

Figures (8)

  • Figure 5.1: Experiment 1.Top-left: the computed lower bounds $\widehat{C}_{\mathrm{DRO}}^{\mathrm{LB}(t)}$ and upper bounds $\widehat{C}_{\mathrm{DRO}}^{\mathrm{LB}(t)}+\hat{\varepsilon}_{\mathrm{sub}}^{(t)}$ plotted against $|\mathrm{supp}(\hat{\tau}^{(t)})|$. Top-right: the computed sub-optimality estimates $\hat{\varepsilon}_{\mathrm{sub}}^{(t)}$ plotted against $|\mathrm{supp}(\hat{\tau}^{(t)})|$. Bottom: the computed approximately optimal scheduled task durations $\hat{{\boldsymbol{x}}}$.
  • Figure 5.2: Experiment 1. The graphs of the computed subdifferential mappings $\partial\tilde{f}_1,\partial\tilde{f}_2,\partial\tilde{f}_{99},\partial\tilde{f}_{100}$.
  • Figure 5.3: Experiment 1. Pair-wise scatter plots of 4 marginals of the computed $\hat{\varepsilon}_{\mathrm{prob}}$-approximate worst-case probability measure $\hat{\mu}$ with respect to $\hat{{\boldsymbol{x}}}$.
  • Figure 5.4: Experiment 2.Left: the computed lower bounds $\widehat{C}_{\mathrm{DRO}}^{\mathrm{LB}(t)}$ and upper bounds $\widehat{C}_{\mathrm{DRO}}^{\mathrm{LB}(t)}+\hat{\varepsilon}_{\mathrm{sub}}^{(t)}$ plotted against $|\mathrm{supp}(\hat{\tau}^{(t)})|$. Right: the computed sub-optimality estimates $\hat{\varepsilon}_{\mathrm{sub}}^{(t)}$ plotted against $|\mathrm{supp}(\hat{\tau}^{(t)})|$.
  • Figure 5.5: Experiment 2. The graphs of the computed subdifferential mappings $\partial\tilde{f}_1,\partial\tilde{f}_2,\partial\tilde{f}_3,\partial\tilde{f}_{4}$.
  • ...and 3 more figures

Theorems & Definitions (35)

  • Lemma 2.2
  • Remark 2.4
  • Example 2.5: Task scheduling chen2018distributionally
  • Example 2.6: Multi-product assembly
  • Example 2.7: Supply chain network design with edge failure
  • Theorem 3.1: Primal and dual formulations of \ref{['eqn:dro']}
  • Corollary 3.2: Property of dual optimizers
  • Remark 3.3
  • Theorem 3.4: Sub-optimality bound for \ref{['eqn:dro-dual']}
  • Corollary 3.5: Sub-optimality bounds for \ref{['eqn:dro-primal']}, \ref{['eqn:dro']}, and the inner maximization problem
  • ...and 25 more