A Simple Counterexample to the Monge Ansatz in Multimarginal Optimal Transport, Convex Geometry of the Set of Kantorovich Plans, and the Frenkel--Kontorova Model

Abstract

It is known from clever mathematical examples [Ca10] that the Monge ansatz may fail in continuous two-marginal optimal transport (alias optimal coupling alias optimal assignment) problems. Here we show that this effect already occurs for finite assignment problems with N = 3 marginals, = 3 'sites', and symmetric pairwise costs, with the values for N and both being optimal. Our counterexample is a transparent consequence of the convex geometry of the set of symmetric Kantorovich plans for N = = 3, which -as we show -possess 22 extreme points, only 7 of which are Monge. These extreme points have a simple physical meaning as irreducible molecular packings, and the example corresponds to finding the minimum energy packing for Frenkel-Kontorova interactions. Our finite example naturally gives rise, by superposition, to a continuous one, where failure of the Monge ansatz manifests itself as nonattainment and formation of 'microstructure'.

Kantorovich problem of optimal transportation of measures: new directions of research

“…Transport problems with many marginals differ from the classical ones in that transport plans are given on a product of more than two (possibly, infinitely many) spaces the projections onto which are fixed. Multimarginal problems were studied by many authors; see the recent papers [19], [22], [51], [62], [67], [71], [74], [77], [81], [110], and [113]- [115], where the reader can find additional references. All the new problems mentioned above can be also set in this situation.…”

Genetic column generation: Fast computation of high-dimensional multi-marginal optimal transport problems

“…Restricting µ to be equal to the prescribed marginal λ * is the classical Monge ansatz from optimal transport theory. But the latter is too restrictive for the validity of Theorem 4.1 when N ≥ 3, even in the case of the uniform marginal λ * = 1 ν=1 δ aν (see [Fri19] for simple counterexamples and [Vög19] for a systematic numerical study).…”