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Quantitative stability analysis for minimax distributionally robust risk optimization

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Abstract

This paper considers distributionally robust formulations of a two stage stochastic programming problem with the objective of minimizing a distortion risk of the minimal cost incurred at the second stage. We carry out a stability analysis by looking into variations of the ambiguity set under the Wasserstein metric, decision spaces at both stages and the support set of the random variables. In the case when the risk measure is risk neutral, the stability result is presented with the variation of the ambiguity set being measured by generic metrics of \(\zeta \)-structure, which provides a unified framework for quantitative stability analysis under various metrics including total variation metric and Kantorovich metric. When the ambiguity set is structured by a \(\zeta \)-ball, we find that the Hausdorff distance between two \(\zeta \)-balls is bounded by the distance of their centers and difference of their radii. The findings allow us to strengthen some recent convergence results on distributionally robust optimization where the center of the Wasserstein ball is constructed by the empirical probability distribution.

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Notes

  1. In some references, it is called Wasserstein metric or Kantorovich–Wasserstein metric, see commentary by Villani [57]. Here we call it Kantorovich metric to distinguish it from Wasserstein metric to be defined later on.

  2. The Dirac-measure is defined by \( \delta _{\xi }(A)=\mathbb {1}_A(\xi )={\left\{ \begin{array}{ll} 1 &{}\quad \text {if }\quad \xi \in A,\\ 0 &{}\quad \text {if }\quad \xi \not \in A. \end{array}\right. }\)

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Acknowledgements

We would like to thank Jie Zhang for an initial proof of Theorem 1 and Shaoyan Guo for careful reading of the paper. We would also like to thank the guest editor and the two anonymous referees for insightful comments which help us significantly strengthen this paper.

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  1. Fakultät für Mathematik, Technische Universität Chemnitz, Chemnitz, Germany

    Alois Pichler

  2. School of Mathematical Sciences, University of Southampton, Southampton, UK

    Huifu Xu

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  1. Alois Pichler
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Correspondence to Alois Pichler.

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Pichler, A., Xu, H. Quantitative stability analysis for minimax distributionally robust risk optimization. Math. Program. 191, 47–77 (2022). https://doi.org/10.1007/s10107-018-1347-4

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