Optimal transport planning with a non linear cost – Guy Bouchitte’ (Universit\\’e de Toulon)


Sala Seminari (Dip. Matematica).


In optimal mass transport theory, many problems can be written in the Monge-Kantorovich form $$ \inf\{ \int_{X\times Y} c(x,y) \, d\gamma \ :\ \gamma\in \Pi(\mu,\ u)\}\ ,\eqno(1) $$ where $\mu,\ u$ 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 $\ u$ respectively (transport plans). Let us recall that if an optimal transport plan $\gamma \in \Pi(\mu, \ u)$ is carried by the graph of a map $T:X\to Y$ i.e. if $$ = \int_X \varphi(x,Tx)\, d\mu \quad,\quad T^\sharp \mu= \ u\ ,$$ then $T$ solves the original Monge problem: \ $ \inf\{ \int_X c(x,Tx) \, d\mu\ :\ T^\sharp \mu= \ u \}.$ \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 $$ = \int_X (\int_X \varphi(x,y) d\gamma^x(y))\, d\mu \ ,$$ and to incorporate in problem $(1)$ an additional cost over $\gamma^x$ as follows $$ \inf \left\{ \int_{X\times X} c(x,y) \, d\gamma + \int_X H(x, \gamma^x) \, d\mu\ :\ \gamma\in \Pi(\mu,\ u)\right\}\ ,\eqno(2) $$ being $H:(x,p) \in X\times \mathcal{P}(X) \to [0,+\infty]$ a suitable non linear function. \bigskip 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). \medskip This is from a joined work with Thierry Champion and J.J. Albert. (

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