Data-Driven Distributionally Robust Chance-Constrained Optimization With Wasserstein Metric

Ji, Ran; Lejeune, Miguel A.

doi:10.2139/ssrn.3201356

Cited by 26 publications

(30 citation statements)

References 19 publications

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“…In addition, for certain loss functions and for P ref = P n being the empirical distribution, the locations of the atoms of π * are also known explicitly. This fact has been exploited to reformulate the distributionally robust chanceconstraints [19,95,46] and to estimate the nonparametric likelihood [63].…”

Section: Dual Reformulation and Examplesmentioning

confidence: 99%

Statistical Analysis of Wasserstein Distributionally Robust Estimators

Blanchet¹,

Murthy²,

Nguyen³

2021

Preprint

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We consider statistical methods which invoke a min-max distributionally robust formulation to extract good out-of-sample performance in data-driven optimization and learning problems. Acknowledging the distributional uncertainty in learning from limited samples, the min-max formulations introduce an adversarial inner player to explore unseen covariate data. The resulting Distributionally Robust Optimization (DRO) formulations, which include Wasserstein DRO formulations (our main focus), are specified using optimal transportation phenomena. Upon describing how these infinite-dimensional min-max problems can be approached via a finite-dimensional dual reformulation, the tutorial moves into its main component, namely, explaining a generic recipe for optimally selecting the size of the adversary's budget. This is achieved by studying the limit behavior of an optimal transport projection formulation arising from an inquiry on the smallest confidence region that includes the unknown population risk minimizer. Incidentally, this systematic prescription coincides with those in specific examples in high-dimensional statistics and results in error bounds that are free from the curse of dimensions. Equipped with this prescription, we present a central limit theorem for the DRO estimator and provide a recipe for constructing compatible confidence regions that are useful for uncertainty quantification. The rest of the tutorial is devoted to insights into the nature of the optimizers selected by the min-max formulations and additional applications of optimal transport projections.

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Section: Dual Reformulation and Examplesmentioning

confidence: 99%

Statistical Analysis of Wasserstein Distributionally Robust Estimators

Blanchet¹,

Murthy²,

Nguyen³

2021

Preprint

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“…Recently, data-driven chance constraints over Wasserstein balls were exactly reformulated as mixed-integer conic constraints [145,146]. Leveraging the strong duality result [147], distributionally robust chance constrained programs with Wasserstein ambiguity set were studied for linear constraints with both right and left hand uncertainty [148], as well as for general nonlinear constraints [149].…”

Section: Data-driven Chance Constrained Programmentioning

confidence: 99%

Optimization under uncertainty in the era of big data and deep learning: When machine learning meets mathematical programming

Ning

You

2019

Computers & Chemical Engineering

262

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This paper reviews recent advances in the field of optimization under uncertainty via a modern data lens, highlights key research challenges and promise of data-driven optimization that organically integrates machine learning and mathematical programming for decision-making under uncertainty, and identifies potential research opportunities. A brief review of classical mathematical programming techniques for hedging against uncertainty is first presented, along with their wide spectrum of applications in Process Systems Engineering. A comprehensive review and classification of the relevant publications on data-driven distributionally robust optimization, data-driven chance constrained program, data-driven robust optimization, and data-driven scenario-based optimization is then presented. This paper also identifies fertile avenues for future research that focuses on a closed-loop data-driven optimization framework, which allows the feedback from mathematical programming to machine learning, as well as scenario-based optimization leveraging the power of deep learning techniques. Perspectives on online learningbased data-driven multistage optimization with a learning-while-optimizing scheme is presented.

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“…Distributionally robust joint chance-constrained programs over Wasserstein ambiguity sets are NP-hard, but they admit an exact mixed-integer conic reformulation if the decision variables and uncertain parameters are separable [24]- [26]. A tractable robust approximation is proposed in [27].…”

Section: Introductionmentioning

confidence: 99%

Energy and reserve dispatch with distributionally robust joint chance constraints

Ordoudis

Nguyên

Kühn

2021

Operations Research Letters

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We develop a two-stage stochastic program for energy and reserve dispatch, which ensures the safe operation of a power system with a high penetration of renewables and a strong interdependence with the natural gas system. Distributionally robust joint chance constraints with Wasserstein ambiguity sets ensure that there is no need for load shedding and renewable spillage with high probability under any distribution compatible with the given statistical data. To make this problem tractable, we solve it in linear decision rules, and we develop a family of conditional value-at-risk (CVaR) approximations for the chance constraints. We show through extensive simulations that the proposed model dominates the corresponding two-stage stochastic program without chance constraints that models the consequences of load shedding and renewable spillage explicitly, both in terms of the mean and variability of the out-of-sample cost.

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Data-Driven Distributionally Robust Chance-Constrained Optimization With Wasserstein Metric

Cited by 26 publications

References 19 publications

Statistical Analysis of Wasserstein Distributionally Robust Estimators

Statistical Analysis of Wasserstein Distributionally Robust Estimators

Optimization under uncertainty in the era of big data and deep learning: When machine learning meets mathematical programming

Energy and reserve dispatch with distributionally robust joint chance constraints

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