A non-linear monotonicity principle and applications to Schr\"odinger type problems

Understand

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 that is applicable to the Schr\"odinger problem and use it to characterize the structure of optimizers for target functionals beyond relative entropy.