Duality for rectified Cost Functions

TL;DR

This paper introduces a rectification of cost functions in optimal transport to ensure duality holds, emphasizing the importance of lower semi-continuity, and shows that this adjustment preserves the primal problem's value.

Contribution

The paper proposes a new notion of rectification for cost functions that guarantees duality in optimal transport, even when the original cost lacks lower semi-continuity.

Findings

Rectification $c_r$ ensures duality for a broader class of cost functions.

The primal value remains unchanged when replacing $c$ with $c_r$.

The rectified cost $c_r$ becomes lower semi-continuous under proper topologies.

Abstract

It is well-known that duality in the Monge-Kantorovich transport problem holds true provided that the cost function $c:X\times Y\to [0,\infty]$ is lower semi-continuous or finitely valued, but it may fail otherwise. We present a suitable notion of \emph{rectificaton} $c_r$ of the cost $c$, so that the Monge-Kantorovich duality holds true replacing $c$ by $c_r$. In particular, passing from $c$ to $c_r$ only changes the value of the primal Monge-Kantorovich problem. Finally, the rectified function $c_r $ is lower semi-continuous as soon as $X$ and $Y$ are endowed with proper topologies, thus emphasizing the role of lower semi-continuity in the duality-theory of optimal transport.