Calculus of Variations and Geometric Measure Theory

Optimal transport planning with a non linear cost

Guy Bouchitté (Université du Sud Toulon-Var)

created by gelli on 22 Mar 2019

3 apr 2019 -- 18:00   [open in google calendar]

Sala Seminari (Dipartimento di Matematica di Pisa)

Abstract.

In optimal mass transport theory, many problems can be written in the Monge-Kantorovich form $ \inf\{ \int_{X\times Y} c(x,y) \, d\gamma \ :\ \gamma\in \Pi(\mu,\nu)\}\ \quad (1) $ where $\mu,\nu$ are given probability measures on $X,Y$ and $c:X\times Y \to [0,+\infty[$ is a cost function. Here the competitors are probability measures $\gamma$ on $X\times Y$ with marginals $\mu$ and $\nu$ respectively (transport plans). Let us recall that if an optimal transport plan $\gamma \in \Pi(\mu, \nu)$ is carried by the graph of a map $T:X\to Y$ i.e. if $ <\gamma, \varphi(x,y)> = \int_X \varphi(x,Tx)\, d\mu \quad,\quad T^\sharp \mu= \nu\ ,$ then $T$ solves the original Monge problem: \ $ \inf\{ \int_X c(x,Tx) \, d\mu\ :\ T^\sharp \mu= \nu \}.$

\bigskip Here we are interested in a different case. Indeed in some applications to economy or in probability theory, it can be interesting to favour optimal plans which are non associated to a single valued transport map $T(x)$. The idea is then to consider, instead of $T(x)$, the family of conditional probabilities $\gamma^x$ such that $ <\gamma, \varphi(x,y)> = \int_X (\int_X \varphi(x,y) d\gamma^x(y))\, d\mu \ ,$ and to incorporate in problem $(1)$ an additional cost over $\gamma^x$ as follows $ \inf \left\{ \int_{X\times X} c(x,y) \, d\gamma + \int_X H(x, \gamma^x) \, d\mu\ :\ \gamma\in \Pi(\mu,\nu)\right\}\ ,\quad (2) $ being $H:(x,p) \in X\times \mathcal{P}(X) \to [0,+\infty]$ a suitable non linear function.

In this talk I will describe some results concerning problem $(2)$ (existence, duality principle, optimality conditions) and focus on specific examples where $X=Y$ and $X$ is a convex compact subset of $\Rbb^d$. We will consider in particular the case where $H(x, p)= - \text{var} (p)$ or where $H(x,\cdot)$ is the indicator of a constraint on the barycenter of $p$ (martingale transport).

This is from a joined work with Thierry Champion and J.J. Alibert.