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：考虑从噪音数据样本中推断出不确定性分布的托盘程序。 托盘程序与分布式热点优化相近, 将最坏的预期成本降到最小值, 也就是与观察到的数据完全相容的一组分布。 在本文中, 模糊性设置可以捕捉一系列概率分布的集合, 这些概率分布与噪音分布的结合仍然集中在以总变异距离为参数的数据样本实证性吵闹分布的球中。 使用规定的模糊性集, 分布式热点优化的解决方案在数据样本数量增长至无限性时会聚集到原始随机程序的解决办法中。 因此, 拟议的分布式热点优化问题在本质上是相同的。 这一点在假定噪音分布式分布式分布式分布式优化以直角为主的情况下得到了证明。 更重要的是, 分配式热点优化问题可以被描绘成可移动的锥形优化问题, 并因此容易发生大规模随机化问题 。