Distributionally Robust Optimization and Its Tractable Approximations

Abstract

In this paper we focus on a linear optimization problem with uncertainties, having expectations in the objective and in the set of constraints. We present a modular framework to obtain an approximate solution to the problem that is distributionally robust and more flexible than the standard technique of using linear rules. Our framework begins by first affinely extending the set of primitive uncertainties to generate new linear decision rules of larger dimensions and is therefore more flexible. Next, we develop new piecewise-linear decision rules that allow a more flexible reformulation of the original problem. The reformulated problem will generally contain terms with expectations on the positive parts of the recourse variables. Finally, we convert the uncertain linear program into a deterministic convex program by constructing distributionally robust bounds on these expectations. These bounds are constructed by first using different pieces of information on the distribution of the underlying uncertainties to develop separate bounds and next integrating them into a combined bound that is better than each of the individual bounds.

Data-driven distributionally robust optimization using the Wasserstein metric: performance guarantees and tractable reformulations

“…At optimality we have λ = max k≤K a k * , which corresponds to the (best) Lipschitz constant of (ξ ) with respect to the norm · . On the other hand, Theorem 6.3 implies that (10) is equivalent to (22) with κ = λ . Thus, Corollary 5.1 and Theorem 6.3 are consistent.…”

“…In this setting, the existing literature has studied three types of ambiguity sets. Moment ambiguity sets contain all distributions that satisfy certain moment constraints, see for example [18,22,51] or the references therein. An attractive alternative is to define the ambiguity set as a ball in the space of probability distributions by using a probability distance function such as the Prohorov metric [20], the Kullback-Leibler divergence [25,27], or the Wasserstein metric [38,52] etc.…”

“…Specifically, there is growing evidence that distributionally robust models with moment ambiguity sets are more tractable than the corresponding stochastic models because the intractable high-dimensional integrals in the objective function are replaced with tractable (generalized) moment problems [18,22,51]. In contrast, distributionally robust models with Wasserstein ambiguity sets are believed to be harder than their stochastic counterparts [36].…”

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].…”

Price of Correlations in Stochastic Optimization

“…The DRSP model was proposed by Scarf as early as the 1950s (Scarf (1958)), but it has not received much attention until recently (Popescu (2007), Delage and Ye (2008), Goh and Sim (2009)). The DRSP model accurately characterizes the challenge of making decisions with limited information about the distribution.…”