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On linear optimization over Wasserstein balls
Mathematical Programming ( IF 2.7 ) Pub Date : 2021-06-17 , DOI: 10.1007/s10107-021-01673-8
Man-Chung Yue , Daniel Kuhn , Wolfram Wiesemann

Wasserstein balls, which contain all probability measures within a pre-specified Wasserstein distance to a reference measure, have recently enjoyed wide popularity in the distributionally robust optimization and machine learning communities to formulate and solve data-driven optimization problems with rigorous statistical guarantees. In this technical note we prove that the Wasserstein ball is weakly compact under mild conditions, and we offer necessary and sufficient conditions for the existence of optimal solutions. We also characterize the sparsity of solutions if the Wasserstein ball is centred at a discrete reference measure. In comparison with the existing literature, which has proved similar results under different conditions, our proofs are self-contained and shorter, yet mathematically rigorous, and our necessary and sufficient conditions for the existence of optimal solutions are easily verifiable in practice.



中文翻译:

关于 Wasserstein 球的线性优化

Wasserstein 球包含与参考度量在预先指定的 Wasserstein 距离内的所有概率度量,最近在分布鲁棒优化和机器学习社区中广受欢迎,以制定和解决具有严格统计保证的数据驱动优化问题。在本技术说明中,我们证明了 Wasserstein 球在温和条件下是弱紧致的,并且我们为最优解的存在提供了充分必要条件。如果 Wasserstein 球以离散参考度量为中心,我们还描述了解的稀疏性。与在不同条件下证明类似结果的现有文献相比,我们的证明是独立的、更短的,但在数学上是严格的,

更新日期:2021-06-18
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