Distributionally Robust Chance-Constrained Programs with Right-Hand Side Uncertainty under Wasserstein Ambiguity

Abstract

We consider exact deterministic mixed-integer programming (MIP) reformulations of distributionally robust chance-constrained programs (DR-CCP) with random right-hand sides over Wasserstein ambiguity sets. The existing MIP formulations are known to have weak continuous relaxation bounds, and, consequently, for hard instances with small radius, or with a large number of scenarios, the branch-and-bound based solution processes suffer from large optimality gaps even after hours of computation time. This significantly hinders the practical application of the DR-CCP paradigm. Motivated by these challenges, we conduct a polyhedral study to strengthen these formulations. We reveal several hidden connections between DR-CCP and its nominal counterpart (the sample average approximation), mixing sets, and robust 0-1 programming. By exploiting these connections in combination, we provide an improved formulation and two classes of valid inequalities for DR-CCP. We test the impact of our results on a stochastic transportation problem numerically. Our experiments demonstrate the effectiveness of our approach; in particular our improved formulation and proposed valid inequalities reduce the overall solution times remarkably. Moreover, this allows us to significantly scale up the number of scenarios that can be handled in such DR-CCP formulations from 100s to 3000.

Consistency of Distributionally Robust Risk- and Chance-Constrained Optimization under Wasserstein Ambiguity Sets

“…We proceed with a contradiction argument to show (16). Recall the assertion that ( 16) holds P ∞ -almost surely.…”

“…that has finite measure under the distribution P ∞ and each element of H violates the limit (16). Here, Σ is some uncountable index set.…”

“…Here, Σ is some uncountable index set. To be more precise, H gives rise to a set of sequences {{F DRRCP N (σ)} N ∈N | σ ∈ Σ} such that each element in this set violates (16). This in turn implies that for each σ ∈ Σ, one can assign a sequence…”

“…Within this paradigm, ambiguity sets defined via the Wasserstein distance (see Section II for the definition) have been shown to have desirable out-of-sample performance and analytical tractability [11]- [13]. Motivated by these attractive features, several recent works have proposed approximations and finite-dimensional reformulations of Wasserstein distributionally robust chance and CVaR constrained programs [13]- [16]. This class of problems have also been studied in the context of statistical learning [17], data-driven control [18], [19], and optimal power flow [20], among others.…”

Distributional Robust Batch Contextual Bandits

“…Our formulation postulates that the future environment-characterized by a joint distribution on the context and all the rewards when taking different actions-is in a Kullback-Leibler neighborhood around the training environment's distribution, thereby allowing for learning a robust policy from training data that is not sensitive to the future environment being the same as the past. Despite the fact that there has been a growing literature (see, e.g, [9,19,31,56,6,26,47,21,61,57,39,14,59,64,41,48,66,46,69,1,68,59,27,14,28,10,23,22,30]) on distributionally robust optimization (DRO)-one that shares the same philosophical underpinning on distributionally robustness as ours-the existing DRO literature has mostly focused on the statistical learning aspects, including supervised learning and feature selection type problems, rather than the decision making aspects. To the best of our knowledge, we provide the first distributionally robust formulation for policy evaluation and learning under bandit feedback, in a general, non-parametric space.…”