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Stochastic programs, where uncertainty distribution must be inferred from noisy data samples, are considered. They 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 observed data. The ambiguity sets capture probability distributions whose convolution with the noise distribution is within a ball centered at the empirical noisy distribution of data samples parameterized by 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 number of the data samples grow to infinity. Therefore, the proposed distributionally/robust optimization problems are asymptotically consistent. The distributionally/robust optimization problems can be cast as tractable optimization problems.

Distributionally Robust Optimization With Noisy Data for Discrete Uncertainties Using Total Variation Distance