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On Linear Optimization over Wasserstein Balls

Man-Chung Yue (manchung.yue***at***polyu.edu.hk)
Daniel Kuhn (daniel.kuhn***at***epfl.ch)
Wolfram Wiesemann (ww***at***imperial.ac.uk)

Abstract: Wasserstein balls, which contain all probability measures within a pre-specified Wasserstein distance to a reference measure, have recently enjoyed wide popularity in the distributionally robust optimization and machine learning communities to formulate and solve data-driven optimization problems with rigorous statistical guarantees. In this technical note we prove that the Wasserstein ball is weakly compact under mild conditions, and we offer necessary and sufficient conditions for the existence of optimal solutions. We also characterize the sparsity of solutions if the Wasserstein ball is centred at a discrete reference measure. In comparison with the existing literature, which has proved similar results under different conditions, our proofs are self-contained and shorter, yet mathematically rigorous, and our necessary and sufficient conditions for the existence of optimal solutions are easily verifiable in practice.

Keywords: Distributionally Robust Optimization, Wasserstein Distance, Data-driven Optimization, Discrete Probability Measures, Weak Compactness

Category 1: Stochastic Programming

Category 2: Robust Optimization


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Entry Submitted: 04/15/2020
Entry Accepted: 04/15/2020
Entry Last Modified: 04/15/2020

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