On linear optimization over Wasserstein balls

Man Chung Yue, Daniel Kuhn, Wolfram Wiesemann

Research output: Journal article publicationJournal articleAcademic researchpeer-review

4 Citations (Scopus)


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.

Original languageEnglish
Number of pages16
JournalMathematical Programming
Publication statusE-pub ahead of print - 17 Jun 2021

ASJC Scopus subject areas

  • Software
  • Mathematics(all)


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