We present an elementary yet general proof of duality for Wasserstein distributionally robust optimization. The duality holds for any arbitrary Kantorovich transport cost, measurable loss function, and nominal probability distribution, provided that an interchangeability principle holds, which is equivalent to certain measurability conditions. To illustrate the broader applicability of our approach, we provide a rigorous treatment of duality results in distributionally robust Markov decision processes and distributionally robust multistage stochastic programming. Furthermore, we extend the result to other problems including infinity-Wasserstein distributionally robust optimization, risk-averse optimization, and globalized distributionally robust counterpart.

翻译：我们给出Wasserstein分布鲁棒优化对偶性的一个初等但通用的证明。该对偶性对任意Kantorovich运输成本、可测损失函数和名义概率分布均成立，前提是满足一个可交换性原则——该原则等价于某些可测性条件。为说明我们方法的更广泛适用性，我们严格处理了分布鲁棒马尔可夫决策过程和分布鲁棒多阶段随机规划中的对偶结果。此外，我们将该结果扩展至其他问题，包括无穷-Wasserstein分布鲁棒优化、风险厌恶优化和全局化分布鲁棒对应问题。