Robust Optimization with Ambiguous Stochastic Constraints Under Mean and Dispersion Information

Abstract

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.

Distributionally robust optimization for sequential decision-making

“…Presenting the statistical-distance-based ambiguity sets in the general format (2.2), we can mitigate the conservativeness by further incorporate the structure information in an intuitive way. For example, we can tailor an ambiguity set that restricts the Wasserstein distance from the ambiguous distribution to the empirical distribution, while at the same time specifies the popular mean absolute deviation from the mean (see, for example, [25]) as follows:…”

Optimization under uncertainty in the era of big data and deep learning: When machine learning meets mathematical programming

“…properties of distributions, such as symmetry, unimodality, multimodality and independence, were further integrated into distributionally robust chance constrained programs leveraging a Choquet representation [115]. Nonlinear extensions of distributionally robust chance constraints were made under the ambiguity sets defined by mean and variance [139], convex moment constraints [140], mean absolute deviation [141], and a mixture of distributions [142].…”

The Convergence of Gallego’s Iterative Method for Distribution-Free Inventory Models

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