TY - JOUR
T1 - On linear optimization over Wasserstein balls
AU - Yue, Man Chung
AU - Kuhn, Daniel
AU - Wiesemann, Wolfram
N1 - Funding Information:
The authors gratefully acknowledge funding from the Swiss National Science Foundation under Grant BSCGI0157733, the UK’s Engineering and Physical Sciences Research Council under Grant EP/R045518/1 and the Hong Kong Research Grants Council under the Grant 25302420.
Publisher Copyright:
© 2021, Springer-Verlag GmbH Germany, part of Springer Nature and Mathematical Optimization Society.
PY - 2021/6/17
Y1 - 2021/6/17
N2 - 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.
AB - 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.
UR - http://www.scopus.com/inward/record.url?scp=85108108253&partnerID=8YFLogxK
U2 - 10.1007/s10107-021-01673-8
DO - 10.1007/s10107-021-01673-8
M3 - Journal article
AN - SCOPUS:85108108253
SN - 0025-5610
JO - Mathematical Programming
JF - Mathematical Programming
ER -