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

TL;DR

This paper extends the concept of cyclical monotonicity from optimal transport to non-linear optimization problems, specifically applying it to Schrödinger problems and non-convex functionals, offering a new characterization method.

Contribution

It introduces a non-linear monotonicity principle applicable to Schrödinger problems, enabling analysis of non-convex functionals beyond traditional convex duality methods.

Findings

Established a monotonicity principle for non-linear optimization.

Characterized optimizers for Schrödinger problems using this principle.

Extended analysis to non-convex target functionals.

Abstract

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.