论文标题
在瓦斯堡球上的线性优化
On Linear Optimization over Wasserstein Balls
论文作者
论文摘要
Wasserstein Balls在预先指定的Wasserstein距离与参考度量的距离内包含所有概率度量，最近在分布强劲的优化和机器学习社区中享有广泛的知名度，以制定和解决具有严格统计保证的数据驱动的优化问题。在此技术说明中，我们证明了在温和条件下，瓦斯恒星球弱紧凑，我们为存在最佳溶液提供了必要的条件。如果Wasserstein Ball以离散的参考度量为中心，我们还表征了解决方案的稀疏性。与现有文献相比，在不同条件下证明了相似的结果，我们的证明是独立的，更短的，但在数学上是严格的，并且在实践中，我们为存在最佳解决方案的必要条件易于证实。
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.
