  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. In
contrast to classical convex duality approaches, a main novelty is that the
monotonicity principle allows to deal also with non-convex functionals.
