Stochastic programs where the uncertainty distribution must be inferred from noisy data samples are considered. The stochastic programs are approximated with distributionally-robust optimizations that minimize the worst-case expected cost over ambiguity sets, i.e., sets of distributions that are sufficiently compatible with the observed data. In this paper, the ambiguity sets capture the set of probability distributions whose convolution with the noise distribution remains within a ball centered at the empirical noisy distribution of data samples parameterized by the total variation distance. Using the prescribed ambiguity set, the solutions of the distributionally-robust optimizations converge to the solutions of the original stochastic programs when the numbers of the data samples grow to infinity. Therefore, the proposed distributionally-robust optimization problems are asymptotically consistent. This is proved under the assumption that the distribution of the noise is uniformly diagonally dominant. More importantly, the distributionally-robust optimization problems can be cast as tractable convex optimization problems and are therefore amenable to large-scale stochastic problems.

翻译：考虑不确定性分布必须从含噪数据样本中推断的随机规划问题。通过分布式鲁棒优化近似求解这些随机规划，该优化在模糊集（即与观测数据充分兼容的分布集合）上最小化最坏情况下的期望成本。本文提出的模糊集捕获了这样一类概率分布：其与噪声分布的卷积仍保持在以数据样本经验含噪分布为中心、由总变差距离参数化的球内。采用该预设模糊集时，随着数据样本数量趋于无穷，分布式鲁棒优化问题的解收敛于原始随机规划问题的解。因此，所提出的分布式鲁棒优化问题具有渐近一致性。这一结论在噪声分布满足均匀对角占优假设的条件下得到证明。更重要的是，该分布式鲁棒优化问题可转化为易处理的凸优化问题，因此适用于大规模随机规划问题。