Multi-marginal Transport Problem Research Articles

Monge–Kantorovich optimal transport through constrictions and flow-rate constraints

We consider the problem to transport resources/mass while abiding by constraints on the flow through constrictions along their path between specified terminal distributions. Constrictions, conceptualized as toll stations at specified points, limit the flow-rate across. We quantify flow-rate constraints via a bound on a sought probability density of the times that mass-elements cross toll stations and cast the transportation scheduling in a Kantorovich-type of formalism. Recent work by our team focused on the existence of Monge maps for similarly constrained transport minimizing average kinetic energy. The present formulation in this paper, besides being substantially more general, is cast as a (generalized) multi-marginal transport problem — a problem of considerable interest in modern-day machine learning literature and motivated extensive computational analyses. An enabling feature of our formalism is the representation of an average quadratic cost on the speed of transport as a convex constraint that involves crossing times.

Marginals with Finite Repulsive Cost

AbstractWe consider a multimarginal transport problem with repulsive cost, where the marginals are all equal to a fixed probability$\unicode[STIX]{x1D70C}\in {\mathcal{P}}(\mathbb{R}^{d})$. We prove that, if the concentration of$\unicode[STIX]{x1D70C}$is less than$1/N$, then the problem has a solution of finite cost. The result is sharp, in the sense that there exists$\unicode[STIX]{x1D70C}$with concentration$1/N$for which the cost is infinite.

Nonlinear systems coupled through multi-marginal transport problems

In this paper, we introduce a dynamical urban planning model. This leads to the study of a system of nonlinear equations coupled through multi-marginal optimal transport problems. A first example consists in solving two equations coupled through the solution to the Monge–Ampère equation. We show that theWasserstein gradient flow theory provides a very good framework to solve these highly nonlinear systems. At the end, a uniqueness result is presented in dimension one based on convexity arguments.

A land of monotone plenty

A fundamental concept in optimal transport is c-cyclical monotonicity: it allows to link the optimality of transport plans to the geometry of their support sets. Recently, related concepts have been successfully applied in the multi-marginal version of the transport problem as well as in the martingale transport problem which arises from model-independent finance. We establish a unifying concept of c-monotonicity / finitistic optimality which describes the geometric structure of optimizers to infinite-dimensional linear programming problems. This allows us to strengthen known results in martingale optimal transport and the infinitely marginal case of the optimal transport problem. If the optimization problem can be formulated as a multi-marginal transport problem our contribution is parallel to a recent result of Zaev.

Convex Relaxation Approaches for Strictly Correlated Density Functional Theory

In this paper, we introduce methods from convex optimization to solve the multimarginal transport type problems that arise in the context of density functional theory. Convex relaxations are used to provide outer approximation to the set of $N$-representable 2-marginals and 3-marginals, which in turn provide lower bounds to the energy. We further propose rounding schemes to obtain upper bound to the energy. Numerical experiments demonstrate a gap of the order of $10^{-3}$ to $10^{-2}$ between the upper and lower bounds. The Kantorovich potential of the multimarginal transport problem is also approximated with a similar accuracy.