From CVaR to Uncertainty Set: Implications in Joint Chance-Constrained Optimization

Abstract

We review and develop different tractable approximations to individual chance constrained problems in robust optimization on a varieties of uncertainty sets and show their interesting connections with bounds on the conditional-value-at-risk (CVaR) measure. We extend the idea to joint chance constrained problems and provide a new formulation that improves upon the standard approach. Our approach builds on a classical worst case bound for order statistics problems and is applicable even if the constraints are correlated. We provide an application of the model on a network resource allocation problem with uncertain demand.

A distributionally robust perspective on uncertainty quantification and chance constrained programming

“…Until recently, such problems were suspected to be generically intractable, and the majority of the literature focused on conservative approximations via Bonferroni's inequality [36], distributionally robust conditional value-at-risk constraints [12,51] and component-wise quasi-concave functions [33]. In the following, we present an exact tractable reformulation of problem (2) for a specific class of ambiguity sets, and we argue that this tractability result is unlikely to extend to more general settings.…”

“…Note that P = ΠzP , where P is defined in (12). Moreover, P can be viewed as the intersection of a moment ambiguity set P n of the form (4) with I = 1 and a structural ambiguity set P s containing all distributions P ∈ P 0 (R P ×R P ) under which ΠzP is symmetric around m. For f > 0, P satisfies the conditions (B), (N), (D), (F) and (S).…”

“…Note that we have P = ΠzP for P defined in (12). If we assume that ∈ (0, 1/2) and apply a coordinate shift, then all conditions of Theorem 9 are met and the chance constraint in (2) is satisfied if and only if…”

“…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,…”

Constructing Uncertainty Sets for Robust Linear Optimization

“…For multiple constraints, we can obviously apply this framework in constraintwise fashion; on the other hand, depending on how we wish to weigh the risk associated with the various constraints, this may or may not be appropriate. Chen et al (2009) Jouini et al (2004) for an effort at extending coherent risk measures to more general vector spaces. Because this is still unresolved, our focus is therefore on a single constraint.…”