Distributionally Robust Convex Optimization

TLDR

Distributionally robust optimization addresses decision making under uncertainty by optimizing against an ambiguity set of probability distributions compatible with prior information. This paper proposes a unifying framework for modeling and solving distributionally robust optimization problems. The framework introduces standardized ambiguity sets comprising all distributions with conic‑representable confidence sets and means on an affine manifold. These expressive ambiguity sets subsume many existing sets, enable characterization via classical and robust statistical indicators, and allow the authors to identify tractability conditions and provide tractable conservative approximations when those conditions are violated.

Abstract

Distributionally robust optimization is a paradigm for decision making under uncertainty where the uncertain problem data are governed by a probability distribution that is itself subject to uncertainty. The distribution is then assumed to belong to an ambiguity set comprising all distributions that are compatible with the decision maker’s prior information. In this paper, we propose a unifying framework for modeling and solving distributionally robust optimization problems. We introduce standardized ambiguity sets that contain all distributions with prescribed conic representable confidence sets and with mean values residing on an affine manifold. These ambiguity sets are highly expressive and encompass many ambiguity sets from the recent literature as special cases. They also allow us to characterize distributional families in terms of several classical and/or robust statistical indicators that have not yet been studied in the context of robust optimization. We determine conditions under which distributionally robust optimization problems based on our standardized ambiguity sets are computationally tractable. We also provide tractable conservative approximations for problems that violate these conditions.