On linear optimization over Wasserstein balls

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