Sufficient conditions for optimality in the Wasserstein space

Establish sufficient optimality conditions for constrained minimization problems over the Wasserstein space P2(Rd). Specifically, develop rigorous sufficient conditions under which a probability measure p* ∈ C ⊂ P2(Rd) is a global or local minimizer of a general functional J: P2(Rd) → R, potentially leveraging geodesic convexity and second-order (Wasserstein) calculus to characterize such sufficiency beyond the necessary conditions provided in this work.

Background

The paper derives general first-order necessary conditions for optimization problems where the decision variable is a probability measure endowed with the Wasserstein metric. While these conditions generalize classical Karush–Kuhn–Tucker and Lagrange conditions to the Wasserstein space, they do not provide sufficiency. The authors indicate that, analogous to Euclidean optimization, sufficiency likely requires additional geometric or second-order structure, such as geodesic convexity or second-order analysis in the Wasserstein space.

The authors highlight preliminary connections to geodesic convexity and second-order Wasserstein calculus, but confirm that establishing comprehensive sufficient conditions remains unresolved and is left for future research.