Bouchitté: Optimal transport planning with a non linear cost

Bouchitté: 
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.
http://cvgmt.sns.it/seminar/683/
When	Wed Apr 3, 2019 4pm – 5pm Coordinated Universal Time
Where	Sala Seminari (Dipartimento di Matematica di Pisa) (map)