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Optimal Transport-Based Distributionally Robust Optimization: Structural Properties and Iterative Schemes

Author

Listed:
  • Jose Blanchet

    (Management Science and Engineering, Stanford University, Stanford, California 94305)

  • Karthyek Murthy

    (Engineering Systems and Design, Singapore University of Technology & Design, Singapore 487372, Singapore)

  • Fan Zhang

    (Management Science and Engineering, Stanford University, Stanford, California 94305)

Abstract

We consider optimal transport-based distributionally robust optimization (DRO) problems with locally strongly convex transport cost functions and affine decision rules. Under conventional convexity assumptions on the underlying loss function, we obtain structural results about the value function, the optimal policy, and the worst-case optimal transport adversarial model. These results expose a rich structure embedded in the DRO problem (e.g., strong convexity even if the non-DRO problem is not strongly convex, a suitable scaling of the Lagrangian for the DRO constraint, etc., which are crucial for the design of efficient algorithms). As a consequence of these results, one can develop efficient optimization procedures that have the same sample and iteration complexity as a natural non-DRO benchmark algorithm, such as stochastic gradient descent.

Suggested Citation

  • Jose Blanchet & Karthyek Murthy & Fan Zhang, 2022. "Optimal Transport-Based Distributionally Robust Optimization: Structural Properties and Iterative Schemes," Mathematics of Operations Research, INFORMS, vol. 47(2), pages 1500-1529, May.
    • Handle: RePEc:inm:ormoor:v:47:y:2022:i:2:p:1500-1529
    DOI: 10.1287/moor.2021.1178
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    References listed on IDEAS

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    1. Paul Milgrom & Ilya Segal, 2002. "Envelope Theorems for Arbitrary Choice Sets," Econometrica, Econometric Society, vol. 70(2), pages 583-601, March.
    2. Viet Anh Nguyen & Daniel Kuhn & Peyman Mohajerin Esfahani, 2018. "Distributionally Robust Inverse Covariance Estimation: The Wasserstein Shrinkage Estimator," Papers 1805.07194, arXiv.org.
    3. Luo, Fengqiao & Mehrotra, Sanjay, 2019. "Decomposition algorithm for distributionally robust optimization using Wasserstein metric with an application to a class of regression models," European Journal of Operational Research, Elsevier, vol. 278(1), pages 20-35.
    4. Jose Blanchet & Karthyek Murthy, 2019. "Quantifying Distributional Model Risk via Optimal Transport," Mathematics of Operations Research, INFORMS, vol. 44(2), pages 565-600, May.
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    Cited by:

    1. Yinchu Zhu & Ilya O. Ryzhov, 2026. "Quantile optimization in semidiscrete optimal transport," Papers 2602.10515, arXiv.org, revised Feb 2026.
    2. Viet Anh Nguyen & Fan Zhang & Shanshan Wang & José Blanchet & Erick Delage & Yinyu Ye, 2025. "Robustifying Conditional Portfolio Decisions via Optimal Transport," Operations Research, INFORMS, vol. 73(5), pages 2801-2829, September.
    3. Shao, Zhiqi & Wang, Ze & Bell, Michael G H & Glenn Geers, D. & Gao, Junbin, 2026. "A distributionally robust chance constraint model to demand-responsive skip planning problem," Transportation Research Part E: Logistics and Transportation Review, Elsevier, vol. 206(C).
    4. Bo Rao & Liu Yang & Jingmin Cai, 2025. "Optimal transport-based distributionally robust optimization with polynomial uncertainty," Journal of Global Optimization, Springer, vol. 93(1), pages 215-240, September.
    5. Hyungki Im & Paul Grigas, 2025. "Stochastic First-Order Algorithms for Constrained Distributionally Robust Optimization," INFORMS Journal on Computing, INFORMS, vol. 37(2), pages 212-229, March.
    6. Ling Liang & Zusen Xu & Kim-Chuan Toh & Jia-Jie Zhu, 2025. "An Inexact Halpern Iteration with Application to Distributionally Robust Optimization," Journal of Optimization Theory and Applications, Springer, vol. 206(3), pages 1-41, September.

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