Data-driven distributionally robust chance-constrained optimization with Wasserstein metric
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Data-driven distributionally robust chance-constrained optimization with Wasserstein metric Ran Ji1
· Miguel A. Lejeune2
Received: 25 June 2019 / Accepted: 31 October 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract We study distributionally robust chance-constrained programming (DRCCP) optimization problems with data-driven Wasserstein ambiguity sets. The proposed algorithmic and reformulation framework applies to all types of distributionally robust chance-constrained optimization problems subjected to individual as well as joint chance constraints, with random right-hand side and technology vector, and under two types of uncertainties, called uncertain probabilities and continuum of realizations. For the uncertain probabilities (UP) case, we provide new mixed-integer linear programming reformulations for DRCCP problems. For the continuum of realizations case with random right-hand side, we propose an exact mixed-integer second-order cone programming (MISOCP) reformulation and a linear programming (LP) outer approximation. For the continuum of realizations (CR) case with random technology vector, we propose two MISOCP and LP outer approximations. We show that all proposed relaxations become exact reformulations when the decision variables are binary or bounded general integers. For DRCCP with individual chance constraint and random right-hand side under both the UP and CR cases, we also propose linear programming reformulations which need the ex-ante derivation of the worst-case value-at-risk via the solution of a finite series of linear programs determined via a bisection-type procedure. We evaluate the scalability and tightness of the proposed MISOCP and (MI)LP formulations on a distributionally robust chance-constrained knapsack problem. Keywords Distributionally robust optimization · Chance-constrained programming · Wasserstein metric · Mixed-integer programming
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Ran Ji [email protected] Miguel A. Lejeune [email protected]
1
Department of Systems Engineering and Operations Research, George Mason University, Fairfax, VA, USA
2
Department of Decision Sciences, George Washington University, Washington, DC, USA
123
Journal of Global Optimization
1 Introduction 1.1 Problem setting Data-driven prescriptive analytics has been receiving growing attention along the emergence of data-centric environments. The continuously increasing availability of data provides great opportunities but also raises concerns revolving around data veracity [13], leading for example to question whether the available data can provide an exact characterization of the sources of uncertainty and their probability distribution. Distributionally robust optimization (see, e.g., [6,22]) has been shown to be a most suitable modeling paradigm to account for such uncertainty. The objective of this paper is to investigate a class of data-driven distributionally robust chance-constrained programming models, in which the stochastic constraints are satisfied with a specified probability level across all possible
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