Robust Optimization with Ambiguous Stochastic Constraints Under Mean and Dispersion Information

Abstract

In this paper we consider ambiguous stochastic constraints under partial information consisting of means and dispersion measures of the underlying random parameters. Whereas the past literature used the variance as the dispersion measure, here we use the mean absolute deviation from the mean (MAD). This makes it possible to use the 1972 result of Ben-Tal and Hochman (BH) in which tight upper and lower bounds on the expectation of a convex function of a random variable are given. First, we use these results to treat ambiguous expected feasibility constraints to obtain exact reformulations for both functions that are convex and concave in the components of the random variable. This approach requires, however, the independence of the random variables and, moreover, may lead to an exponential number of terms in the resulting robust counterparts. We then show how upper bounds can be constructed that alleviate the independence restriction, and require only a linear number of terms, by exploiting models in which random variables are linearly aggregated. Moreover, using the BH bounds we derive three new safe tractable approximations of chance constraints of increasing computational complexity and quality. In a numerical study, we demonstrate the efficiency of our methods in solving stochastic optimization problems under mean-MAD ambiguity. The electronic companion is available at https://doi.org/10.1287/opre.2017.1688 .