Multi-marginal optimal transport: Theory and applications^∗

Department of Mathematical and Statistical Sciences, 632 CAB, University of Alberta, Edmonton, Alberta, T6G 2G1, Canada.
pass@ualberta.ca.

Received: 13 March 2015

Abstract

Over the past five years, multi-marginal optimal transport, a generalization of the well known optimal transport problem of Monge and Kantorovich, has begun to attract considerable attention, due in part to a wide variety of emerging applications. Here, we survey this problem, addressing fundamental theoretical questions including the uniqueness and structure of solutions. The answers to these questions uncover a surprising divergence from the classical two marginal setting, and reflect a delicate dependence on the cost function, which we then illustrate with a series of examples. We go on to describe some applications of the multi-marginal optimal transport problem, focusing primarily on matching in economics and density functional theory in physics.