Robust Solutions of Optimization Problems Affected by Uncertain Probabilities

Abstract

In this paper we focus on robust linear optimization problems with uncertainty regions defined by φ-divergences (for example, chi-squared, Hellinger, Kullback-Leibler). We show how uncertainty regions based on φ-divergences arise in a natural way as confidence sets if the uncertain parameters contain elements of a probability vector. Such problems frequently occur in, for example, optimization problems in inventory control or finance that involve terms containing moments of random variables, expected utility, etc. We show that the robust counterpart of a linear optimization problem with φ-divergence uncertainty is tractable for most of the choices of φ typically considered in the literature. We extend the results to problems that are nonlinear in the optimization variables. Several applications, including an asset pricing example and a numerical multi-item newsvendor example, illustrate the relevance of the proposed approach.

A distributionally robust perspective on uncertainty quantification and chance constrained programming

“…Note that (26) is a system of linear constraints that scales polynomially in the description of problem (2). The next result shows that the restriction to Markov ambiguity sets in Theorem 7 is necessary.…”

“…We now study chance constrained programs of the form (2), where the safety of the underlying system can be actively enhanced by adjusting the design decisions x ∈ R N . It turns out that the tractability of (2) is intimately related to the number of rows J of the technology matrix S(x) and the right-hand side vector t(x).…”

“…(ii) The structure of P should facilitate a tractable reformulation (or at least a tractable conservative approximation) of the uncertainty quantification problem (1) and the chance constrained program (2). (iii) Among all ambiguity sets satisfying the properties (i) and (ii), P should be chosen as small as possible in the sense of set inclusion.…”

“…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,…”

“…An intimately related secondary objective is to explore the boundaries of tractability. It is thus natural to focus attention on linear safety constraints in the probabilistic expressions of (1) and (2). Indeed, the uncertainty quantification problem (1) becomes intractable already in the presence a single convex quadratic safety constraint, even if the underlying ambiguity set contains all distributions supported on a polytope.…”