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Shortfall-Based Wasserstein Distributionally Robust Optimization

Author

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  • Ruoxuan Li

    (Department of Statistics and Finance, University of Science and Technology of China, Hefei 230052, China)

  • Wenhua Lv

    (School of Mathematics and Finance, Chuzhou University, Chuzhou 239000, China)

  • Tiantian Mao

    (Department of Statistics and Finance, University of Science and Technology of China, Hefei 230052, China)

Abstract

In this paper, we study a distributionally robust optimization (DRO) problem with affine decision rules. In particular, we construct an ambiguity set based on a new family of Wasserstein metrics, shortfall–Wasserstein metrics, which apply normalized utility-based shortfall risk measures to summarize the transportation cost random variables. In this paper, we demonstrate that the multi-dimensional shortfall–Wasserstein ball can be affinely projected onto a one-dimensional one. A noteworthy result of this reformulation is that our program benefits from finite sample guarantee without a dependence on the dimension of the nominal distribution. This distributionally robust optimization problem also has computational tractability, and we provide a dual formulation and verify the strong duality that enables a direct and concise reformulation of this problem. Our results offer a new DRO framework that can be applied in numerous contexts such as regression and portfolio optimization.

Suggested Citation

  • Ruoxuan Li & Wenhua Lv & Tiantian Mao, 2023. "Shortfall-Based Wasserstein Distributionally Robust Optimization," Mathematics, MDPI, vol. 11(4), pages 1-25, February.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:4:p:849-:d:1060638
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    References listed on IDEAS

    as
    1. Tiantian Mao & Jun Cai, 2018. "Risk measures based on behavioural economics theory," Finance and Stochastics, Springer, vol. 22(2), pages 367-393, April.
    2. Wolfram Wiesemann & Daniel Kuhn & Melvyn Sim, 2014. "Distributionally Robust Convex Optimization," Operations Research, INFORMS, vol. 62(6), pages 1358-1376, December.
    3. Hans Föllmer & Alexander Schied, 2002. "Convex measures of risk and trading constraints," Finance and Stochastics, Springer, vol. 6(4), pages 429-447.
    4. Tiantian Mao & Ruodu Wang & Qinyu Wu, 2022. "Model Aggregation for Risk Evaluation and Robust Optimization," Papers 2201.06370, arXiv.org, revised Oct 2023.
    5. Jose Blanchet & Karthyek Murthy, 2019. "Quantifying Distributional Model Risk via Optimal Transport," Mathematics of Operations Research, INFORMS, vol. 44(2), pages 565-600, May.
    6. Ioana Popescu, 2007. "Robust Mean-Covariance Solutions for Stochastic Optimization," Operations Research, INFORMS, vol. 55(1), pages 98-112, February.
    7. Philippe Artzner & Freddy Delbaen & Jean‐Marc Eber & David Heath, 1999. "Coherent Measures of Risk," Mathematical Finance, Wiley Blackwell, vol. 9(3), pages 203-228, July.
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