Distributionally Robust Convex Optimization

Distributionally robust optimization is a paradigm for decision-making under uncertaintywhere the uncertain problem data is governed by a probability distribution that is itself subjectto uncertainty. The distribution is then assumed to belong to an ambiguity set comprising alldistributions that are compatible with the decision maker’s prior information. In this paper,we propose a unifying framework for modeling and solving distributionally robust optimizationproblems. We introduce standardized ambiguity sets that contain all distributions with pre-scribed 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 recentliterature as special cases. They also allow us to characterize distributional families in terms ofseveral classical and/or robust statistical indicators that have not yet been studied in the contextof robust optimization. We determine conditions under which distributionally robust optimiza-tion problems based on our standardized ambiguity sets are computationally tractable. We alsoprovide tractable conservative approximations for problems that violate these conditions.