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On linear optimization over Wasserstein balls
- Source :
- Mathematical Programming. 195:1107-1122
- Publication Year :
- 2021
- Publisher :
- Springer Science and Business Media LLC, 2021.
-
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.
- Subjects :
- FOS: Computer and information sciences
90C05
Computer Science - Machine Learning
Technology
Operations Research
Optimization problem
Wasserstein metric
Linear programming
General Mathematics
Mathematics, Applied
0211 other engineering and technologies
010103 numerical & computational mathematics
02 engineering and technology
01 natural sciences
Measure (mathematics)
90C25
Machine Learning (cs.LG)
0102 Applied Mathematics
FOS: Mathematics
Applied mathematics
0101 mathematics
Mathematics - Optimization and Control
0802 Computation Theory and Mathematics
Probability measure
Mathematics
90C17
Infinite-dimensional optimization
Science & Technology
021103 operations research
Operations Research & Management Science
0103 Numerical and Computational Mathematics
Robust optimization
Linear optimization
Computer Science, Software Engineering
Optimization and Control (math.OC)
Physical Sciences
Computer Science
Ball (bearing)
Software
Subjects
Details
- ISSN :
- 14364646 and 00255610
- Volume :
- 195
- Database :
- OpenAIRE
- Journal :
- Mathematical Programming
- Accession number :
- edsair.doi.dedup.....a0acde1ac369cdfc76aff27655f90953