Distributionally Robust Convex Optimization

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.

K-Adaptability in Two-Stage Robust Binary Programming

“…For a general introduction to distributionally robust optimization we refer to [7,11,22] and the references therein. To the best of our knowledge, so far only the literature on persistency in combinatorial optimization has addressed distributionally robust two-stage integer programs [16].…”

A distributionally robust perspective on uncertainty quantification and chance constrained programming

“…Ambiguity sets of special interest include the Markov ambiguity set containing all distributions with known mean and support [48], the Chebyshev ambiguity set containing all distributions with known bounds on the first and second-order moments [12,14,22,31,39,46,49,51,52], the Gauss ambiguity set containing all unimodal distributions from within the Chebyshev ambiguity set [38,41], various generalized Chebyshev ambiguity sets that specify asymmetric moments [12,13,35], higher-order moments [7,30,45] or marginal moments [17,18], the median-absolute deviation ambiguity set containing all symmetric distributions with known median and mean absolute deviation [24], the Huber ambiguity set containing all distributions with known upper bound on the expected Huber loss function [15,48], the Wasserstein ambiguity set containing all distributions that are close to the empirical distribution with respect to the Wasserstein metric [19,34,40], the KullbackLeibler divergence ambiguity set and likelihood ratio ambiguity set [10,26,27,31,47] containing all distributions that are sufficiently likely to have generated a given data set, the Hoeffding ambiguity set containing all component-wise independent distributions with a box support [3,8,10], the Bernstein ambiguity set containing all distributions from within the Hoeffding ambiguity set subject to marginal moment bounds [36], several φ-divergence-based ambiguity sets [2,…”

“…In [48] it has been shown that most moment-based ambiguity sets emerge as special cases of a canonical ambiguity set that contains all distributions under which the probabilities of some conic-representable confidence sets fall between prescribed upper and lower bounds, and the mean values of the uncertain parameters satisfy a linear equality constraint. While [48] describes methods for computing worst-case expectations of biconvex loss functions, the focus of the present paper is to compute worst-case probabilities, that is, worst-case expectations of discontinuous indicator functions.…”

“…While [48] describes methods for computing worst-case expectations of biconvex loss functions, the focus of the present paper is to compute worst-case probabilities, that is, worst-case expectations of discontinuous indicator functions. Moreover, while [48] focuses exclusively on moment-based ambiguity sets, the present paper investigates a much richer class of ambiguity sets characterized both in terms of moment constraints and structural information such as symmetry, unimodality, multimodality, independence patterns etc. In particular, we also show that several classical inequalities of probability theory as well as their multidimensional generalizations emerge as special cases of our unified framework.…”

Routing Optimization Under Uncertainty

“…Therefore, this distributionally robust optimization framework is potentially as tractable as robust optimization and has the benefit of being less conservative. We refer interested readers to Wiesemann et al (2014) for more discussion on it.…”