  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.
