We study stochastic optimization problems with chance and risk constraints, where in the latter, risk is quantified in terms of the conditional value-at-risk (CVaR). We consider the distributionally robust versions of these problems, where the constraints are required to hold for a family of distributions constructed from the observed realizations of the uncertainty via the Wasserstein distance. Our main results establish that if the samples are drawn independently from an underlying distribution and the problems satisfy suitable technical assumptions, then the optimal value and optimizers of the distributionally robust versions of these problems converge to the respective quantities of the original problems, as the sample size increases.

中文翻译：

---

Wasserstein 模糊集下分布鲁棒风险和机会约束优化的一致性


我们研究具有机会和风险约束的随机优化问题，其中风险是根据条件风险价值（CVaR）来量化的。我们考虑这些问题的分布稳健版本，其中约束需要适用于通过 Wasserstein 距离观察到的不确定性实现构建的分布族。我们的主要结果表明，如果样本是独立于基础分布抽取的，并且问题满足适当的技术假设，那么这些问题的分布稳健版本的最优值和优化器会收敛到原始问题的相应数量，如样本尺寸增加。