On Linear Optimization over Wasserstein Balls
This work addresses distributionally robust optimization and machine learning problems by offering rigorous mathematical foundations, though it is incremental as it builds on existing literature with shorter proofs and more practical conditions.
The paper tackles the problem of linear optimization over Wasserstein balls by proving weak compactness under mild conditions and providing necessary and sufficient conditions for optimal solutions, with results including self-contained proofs and easily verifiable conditions.
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.