A non‐linear monotonicity principle and applications to Schrödinger‐type problems

A basic idea in optimal transport is that optimizers can be characterized through a geometric property of their support sets called cyclical monotonicity. In recent years, similar monotonicity principles have found applications in other fields where infinite-dimensional linear optimization problems play an important role. In this note, we observe how this approach can be transferred to non-linear optimization problems. Specifically we establish a monotonicity principle is applicable to the Schrodinger problem and use it to characterize the structure of optimizers for target functionals beyond relative entropy. In contrast to classical convex duality approaches, a main novelty is that the monotonicity principle allows to deal also with non-convex functionals.