The Fundamental Theorem of Weak Optimal Transport

The fundamental theorem of classical optimal transport establishes strong duality and characterizes optimizers through a complementary slackness condition. Milestones such as Brenier’s theorem and the Kantorovich-Rubinstein formula are direct consequences. In this paper, we generalize this result to non-linear cost functions, thereby establishing a fundamental theorem for the weak optimal transport problem introduced by Gozlan, Roberto, Samson, and Tetali. As applications we provide concise derivations of the Brenier–Strassen theorem, the convex Kantorovich–Rubinstein formula and the structure theorem of entropic optimal transport. We also extend Strassen’s theorem in the direction of Gangbo–McCann’s transport problem for convex costs. Moreover, we determine the optimizers for a new family of transport problems which contains the Brenier–Strassen, the martingale Benamou–Brenier and the entropic martingale transport problem as extreme cases.