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

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 nonlinear optimization problems. Specifically we establish a monotonicity principle is applicable to the Schrödinger 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.M S C 2 0 2 0 46N30 (primary), 49Q22, 60F10 (secondary) INTRODUCTION AND MAIN RESULTS Motivation from optimal transportGiven probabilities 𝜇 and 𝜈 on Polish spaces 𝑋 and 𝑌, and a cost function 𝑐 ∶ 𝑋 × 𝑌 → ℝ + , the Monge-Kantorovich problem is to find a cost-minimizing transport plan. More precisely, writing cpl(𝜇, 𝜈) for the set of all couplings (namely, measures) on 𝑋 × 𝑌 with 𝑋-marginal 𝜇 and 𝑌-marginal 𝜈, the problem is to find inf { ∫ 𝑐 𝑑ℙ ∶ ℙ ∈ cpl(𝜇, 𝜈) } (OT)