Kantorovich-Rubinstein duality theory for the Hessian

TL;DR

This paper extends Kantorovich-Rubinstein duality to a setting involving Hessian constraints, linking optimal transport with three-point plans and applications in mechanics.

Contribution

It introduces a duality theory for a class of functions with Hessian bounds, involving three-point plans and convex order, expanding optimal transport applications.

Findings

Established duality with Hessian constraints

Expressed solutions as rank-one tensor measures

Applied to optimal grillage configuration in mechanics

Abstract

The classical Kantorovich-Rubinstein duality theorem establishes a significant connection between Monge optimal transport and maximization of a linear form on the set of 1-Lipschitz functions. This result has been widely used in various research areas. In particular, it unlocks the optimal transport methods in some of the optimal design problems. This paper puts forth a similar theory when the linear form is maximized over $C^{1,1}$ functions whose Hessian lies between minus and plus identity matrix. The problem will be identified as the dual of a specific optimal transport formulation that involves three-point plans. The first two marginals are fixed, while the third must dominate the other two in the sense of convex order. The existence of optimal plans allows to express solutions of the underlying Beckmann problem as a combination of rank-one tensor measures supported on a graph. In the context of two-dimensional mechanics, this graph encodes the optimal configuration of a grillage that transfers a given load system.