\documentclass{beamer}
\usepackage[latin1]{inputenc}
\usepackage{graphicx}
\usepackage{amscd}

\usepackage{beamerthemeshadow}

\newcommand{\Card}{\operatorname{Card}}

\title{Interval analysis and Optimal Transport} \subtitle{}
\author[Nicolas Delanoue]{Nicolas Delanoue - Mehdi Lhommeau - Philippe Lucidarme \\
  \tiny{LARIS - Universite d'Angers - France}} \date{ }
\institute{SWIM 2014  \\
 7th Small Workshop on Interval Methods\\
  University Main Building, Uppsala, Sweden \\
  \url{http://www.math.uu.se/swim2014/} }
\newcommand{\R}{\mathbb{R}} \newcommand{\IR}{\mathbb{IR}}
\newcommand{\Sol}{\mathbb{S}} \newcommand{\C}{\mathcal{C}}
\newcommand{\Pav}{\mathcal{P}}


\definecolor{Mygreen}{rgb}{0,0.5,0}
\definecolor{Myyellow}{rgb}{0.7,0.7,0.0}

\begin{document}

\frame{ 
\titlepage
}

% Expliquer pourquoi j'ai fais ca et les extensions
% préparer la preuve de la relaxation du dual

\frame{ \frametitle{Outline} \tableofcontents }


\section{Introduction to Optimal transport}

\subsection{Transportation}

\frame{
\begin{block}{Example with books}
 \includegraphics[height = 50 pt]{image/livre3.jpg} \\
\end{block}

\begin{block}{Example in the discrete case}<2->
 \includegraphics{image/books.pdf} 
\end{block}
}


\frame{
\begin{block}{Example in the discrete case}
 \includegraphics{image/books.pdf} 
\end{block}
\begin{block}{}<2->
 \includegraphics{image/books2.pdf} 
\end{block}
}


\frame{
\begin{block}{Example in the discrete case}
 \includegraphics{image/books_color.pdf} 
\end{block}
\begin{block}{}
 \includegraphics{image/books2.pdf} 
\end{block}
}


\frame{
\begin{block}{Example in the discrete case}
 \includegraphics{image/books_color.pdf} 
\end{block}
\begin{block}{}
 \includegraphics{image/books2_color.pdf} 
\end{block}
}

\frame{
\begin{block}{Example in the discrete case}
\begin{tabular}{cc}
 \includegraphics[width = 150 px]{image/books.pdf}  &  \includegraphics[width = 150 px]{image/books2.pdf}    \\
 $\mu = (2, 1, 4 )$   &  $\nu = (4, 2, 1 )$  \\
\end{tabular}
\end{block}

\begin{block}{Transportation}
\[ 
\begin{array}{c|ccc}
  & 4 & 2 &  1 \\
\hline
2 &  &  &   \\
1 &  &  &   \\
4 &  &  &   
\end{array} \; \; \;  \; \; \;  \; \; \; \; \; \;  \; \; \;  \; \; \; \; \; \;  \; \; \;  \; \; \; \; \; \;  \; \; \;  \; \; \; \; \; \;  \; \; \; 
\]
\end{block}
}


\frame{
\begin{block}{Example in the discrete case}
\begin{tabular}{cc}
 \includegraphics[width = 150 px]{image/books_color.pdf}  &  \includegraphics[width = 150 px]{image/books2_color.pdf}    \\
 $\mu = (2, 1, 4 )$   &  $\nu = (4, 2, 1 )$  \\
\end{tabular}
\end{block}

\begin{block}{Transportation}
\[ 
\begin{array}{c|ccc}
  & 4 & 2 &  1 \\
\hline
2 &  &  &   \\
1 &  &  &   \\
4 &  &  &   
\end{array} \; \; \;  \; \; \;  \; \; \; \; \; \;  \; \; \;  \; \; \; \; \; \;  \; \; \;  \; \; \; \; \; \;  \; \; \;  \; \; \; \; \; \;  \; \; \; 
\]
\end{block}
}


\frame{
\begin{block}{Example in the discrete case}
\begin{tabular}{cc}
 \includegraphics[width = 150 px]{image/books_color.pdf}  &  \includegraphics[width = 150 px]{image/books2_color.pdf}    \\
 $\mu = (2, 1, 4 )$   &  $\nu = (4, 2, 1 )$  \\
\end{tabular}
\end{block}

\begin{block}{A plan transference $\pi$}
\[ 
\begin{array}{c|ccc}
  & 4 & 2 &  1 \\
\hline
2 & 2 & 0 &  0 \\
1 & 1 & 0 &  0 \\
4 & 1 & 2 &  1 
\end{array}, \; \; \;  \; \; \;  \; \; \; 
\pi = 
\left(
\begin{array}{ccc}
 2 & 0 &  0 \\
 1 & 0 &  0 \\
 1 & 2 &  1 
\end{array}
\right)
\]

\end{block}
}





\frame{

\begin{block}{Plan transference problem}<1->
\[
\begin{array}{c|ccc}
  & 4 & 2 &  1 \\
\hline
2 & . & . &  . \\
1 & . & . &  . \\
4 & . & . &  . 
\end{array}
\]
\end{block}


\begin{block}{Solutions}<2->
\[
\pi = 
\begin{array}{c|ccc}
  & 4 & 2 &  1 \\
\hline
2 & 2 & 0 & 0 \\
1 & 1 & 0 & 0 \\
4 & 1 & 2 & 1 
\end{array}, 
\; 
\; 
\; 
\; 
\tilde{\pi} = 
\begin{array}{c|ccc}
  & 4 & 2 &  1 \\
\hline
2 & 1 & 0 & 1 \\
1 & 1 & 0 & 0 \\
4 & 2 & 2 & 0 
\end{array}. 
\]
\end{block}
}


\frame{
\begin{block}{Definition - Transference plan}
  A transference plan (or a transportation) $\pi$ is a measure on the
  product space $X \times Y$ such that 
  \[ \left\{
    \begin{array}{l}
      \pi( A \times Y) = \mu(A), \\
      \pi( X \times B) = \nu(B).
    \end{array} 
  \right. \]
  all measurable subsets $A$ of $X$ and $B$ of $Y$. 
\end{block}

\begin{block}{In the discrete case}<2->
  \[ \left\{
    \begin{array}{l}
\forall i, \; \sum_j{\pi_{ij} = \mu_i},  \\
\forall j, \; \sum_j{\pi_{ij} = \nu_j}.  \\
    \end{array} 
  \right. \]
\end{block}
}


\frame{
\begin{block}{Comparing two plan transferences}
\begin{tabular}{c|c}
Transportation $\pi$ & Transportation  $\tilde{\pi}$  \\
 & \\
\includegraphics[width = 140 px]{image/books_color.pdf} &  \includegraphics[width = 140 px]{image/books_color.pdf} \\
& \\
\includegraphics[width = 140 px]{image/books2_color.pdf} &  \includegraphics[width = 140 px]{image/books3_color.pdf}  \\
\tiny{$I(\pi) = \textcolor{Mygreen}{0 + 0} + \textcolor{red}{1} +  \textcolor{Myyellow}{2 +  1  + 1 + 0} = 5$}
& 
\tiny{$I(\tilde{\pi}) = \textcolor{Mygreen}{0 + 2} + \textcolor{red}{1} +  \textcolor{Myyellow}{2 +  2  + 1 + 1} = 9$}
\end{tabular}
\end{block}
}


\subsection{Optimal transport}
\frame{
\begin{block}{In the discrete case}
\begin{equation}
\begin{aligned}
& \underset{\pi \in \mathbb{R}^{n} \otimes \mathbb{R}^{m}}{\text{min}}
& & \sum_{i,j} c_{ij} \pi_{ij}  \\
& \text{subject to} 
& & \forall i, \; \sum_j{\pi_{ij} = \mu_i},  \\ 
& & & \forall j, \; \sum_j{\pi_{ij} = \nu_j}. 
\end{aligned}
\end{equation}
where $c_{ij}$ are non negative real numbers which tells how much it
costs to transport one unit of mass from location $i$ to location $j$.
\end{block}
}

\frame{
\begin{block}{Kantorovich formulation}
The \emph{optimal transportation cost} between $\mu$ and $\nu$ is the value :
\begin{equation}
\begin{aligned}
\mathcal{T}_c(\mu,\nu) = & \underset{\pi \in \mathcal{B} ( X \times Y )}{\text{inf}}
& & \int_{X \times Y} c(x,y) \mathrm{d} \pi(x,y)  \\
& \text{subject to} 
& &  \pi_X = \mu, 
\\ & & &\pi_Y = \nu \\
\end{aligned}
\label{transport}
\end{equation}
\end{block}

The optimal $\pi$'s, i.e. those such that $I(\pi) =
\mathcal{T}_c(\mu,\nu) $, if they exist, will be called \emph{optimal
  transference plans}.  

\begin{center}
 \includegraphics[height = 70 pt]{image/kanto.jpg} 
\end{center}
}


\frame{
\begin{block}{Remark}
The \emph{optimal transportation problem} is an infinite dimensional linear programming problem. \\
i.e. $I$ is a linear cost function, and constraints are linear.
\end{block}
}


\subsection{Some known results}

\frame{
\begin{block}{}
\begin{enumerate}
 \item<1-> $c = \| x-y \|^p$, $p>1$ , the strict convexity of $c$
   guarantees that, if $\mu$, $\nu$ are absolutely continuous with
   respect to Lebesgue measure, then there is a unique solution to the
   Kantorovich problem.
 \item<2-> $c = \| x-y \|^2$, optimal transference plans are the (restrictions of) gradients of convex functions.
 \item<3-> many others in 
\begin{center}
 \includegraphics[height = 70 pt]{image/villani.jpg} \\
\emph{Topics in Optimal Transportation}, Cédric Villani, AMS (2003)
\end{center}
\end{enumerate}
\end{block}
}


\section{A lower bound of the optimal value}

\subsection{Finite dimensional relaxation}
\frame{
\begin{small}
\begin{block}{Proposition - Relaxation}
  Let $\mu$ and $\nu$ (with support $X$ and $Y$) be absolutely
  continuous measures with respect to Lebesgue measure. If $\{X_i\}_i$
  and $\{Y_i\}_i$ be finite pavings of $X$ and $Y$. Suppose that
  $\mu(X_i) \in [\underline{\mu}_i, \overline{\mu}_i]$, $\nu(Y_j) \in [\underline{\nu}_j, \overline{\nu}_j]$,
  and $\forall x,y \in X_i\times Y_j, \underline{c}_{ij} \leq c(x,y) $,
\end{block}

\begin{block}{}<2->
\begin{equation*}
\begin{aligned}
\underline{\mathcal{T}} = & \underset{ \pi_{ij} \in \mathbb{R}^{n} \otimes \mathbb{R}^{m}  }{\text{min}}
& & \sum_{i,j} \underline{c}_{ij} \pi_{ij}   \\
& \text{subject to} 
& & \forall i, \;  \underline{\mu}_i \leq \sum_{j} \pi_{ij} \leq \overline{\mu}_i , \\ 
& & & \forall j, \; \underline{\nu}_j  \leq \sum_{i} \pi_{ij} \leq \overline{\nu}_j , \\
& & & \forall i, \forall j, \; \pi_{ij} \geq 0.
\end{aligned}
\end{equation*}
\end{block}


\begin{block}{}<3->
 \[ \textrm{ then } \;\;\;\;\;\;\; \underline{\mathcal{T}} \leq \mathcal{T}_c(\mu,\nu). \]
\end{block}
\end{small}
}


\frame{
\begin{block}{Spatial discretization}
\begin{tabular}{cc}
 \includegraphics[width = 140 px]{image/example_0.pdf}  &  \includegraphics[width = 140 px]{image/example_1.pdf}    \\
 $\mu$   &  $\nu $  \\
\end{tabular}
\end{block}
}


\frame{
\begin{block}{Spatial discretization}
\begin{center}
 \includegraphics[width = 190 px]{image/example_2.pdf}
\end{center}
\end{block}
}


\frame{
\begin{block}{Spatial discretization}
\begin{center}
 \includegraphics[width = 190 px]{image/example_3.pdf}
\end{center}
\end{block}
}


 \frame{
 \begin{block}{Enclosing}
 If $\mu = f(x) \mathrm{d} x$, and $[f]$ an inclusion function for $f$ then 
 \[   \int_X f(x) \mathrm{d} x  \in \sum_i [f](X_i) \lambda(X_i)  \] 
 \begin{tabular}{ccc}
 \includegraphics[width = 85 px]{image/integrale2.pdf} &  \includegraphics[width = 85 px]{image/integrale.pdf}  & \includegraphics[width = 85 px]{image/integrale3.pdf}    \\
 $\underline{\mu}_i$ & $\mu(X)$  & $\overline{\mu}_i$ \\
& & \\
 $\sum_i \underline{f}(X_i) \lambda(X_i) \leq $ & $\int_X f(x) \mathrm{d}x$ & $\leq \sum_i \overline{f}(X_i) \lambda(X_i) $
\end{tabular}
 \end{block}
 }


\frame{
\begin{block}{Proof}
Let $\{X_i\}$, $\{Y_j\}$ be a pavings, let $\pi_{ij} = \pi(X_i \times Y_j)$ then \\
$    \forall \pi, \exists \xi_{ij} \in X_i\times Y_j$,
\begin{equation}
 \sum_{i,j} c(\xi_{ij}) \pi_{ij} =  \int_{X \times Y} c(x,y) \mathrm{d} \pi(x,y)
\end{equation}
\end{block}

\begin{block}{}<2->
Since $\underline{c}_{ij} \leq c(\xi_{ij})$ and $\pi_{ij} \geq 0$,  then \\
$    \forall \pi,$
\begin{equation}
  \sum_{i,j} \underline{c}_{ij} \pi_{ij} \leq  \int_{X \times Y} c(x,y) \mathrm{d} \pi(x,y) 
\end{equation}
\end{block}
}

\frame{
\begin{small}
\begin{block}{Proposition - Relaxation}
  Let $\mu$ and $\nu$ (with support $X$ and $Y$) be absolutely
  continuous measures with respect to Lebesgue measure. If $\{X_i\}_i$
  and $\{Y_i\}_i$ be finite pavings of $X$ and $Y$. Suppose that
  $\mu(X_i) \in [\underline{\mu}_i, \overline{\mu}_i]$, $\nu(Y_j) \in [\underline{\nu}_j, \overline{\nu}_j]$,
  and $\forall x,y \in X_i\times Y_j, \underline{c}_{ij} \leq c(x,y) $,
\end{block}

\begin{block}{}
\begin{equation*}
\begin{aligned}
\mathcal{K} = & \underset{ \pi_{ij} \in \mathbb{R}^{n} \otimes \mathbb{R}^{m}  }{\text{min}}
& & \sum_{i,j} \underline{c}_{ij} \pi_{ij}   \\
& \text{subject to} 
& & \forall i, \;  \mu_i = \sum_{j} \pi_{ij} = \mu_i , \\ 
& & & \forall j, \; \nu_j  = \sum_{i} \pi_{ij} = \nu_j , \\
& & & \forall i, \forall j, \; \pi_{ij} \geq 0.
\end{aligned}
\end{equation*}
\end{block}


\begin{block}{}
 \[ \textrm{ then } \;\;\;\;\;\;\; \mathcal{K} \leq \mathcal{T}_c(\mu,\nu). \]
\end{block}
\end{small}
}



\frame{
\begin{small}
\begin{block}{Proposition - Relaxation}
  Let $\mu$ and $\nu$ (with support $X$ and $Y$) be absolutely
  continuous measures with respect to Lebesgue measure. If $\{X_i\}_i$
  and $\{Y_i\}_i$ be finite pavings of $X$ and $Y$. Suppose that
  $\mu(X_i) \in [\underline{\mu}_i, \overline{\mu}_i]$, $\nu(Y_j) \in [\underline{\nu}_j, \overline{\nu}_j]$,
  and $\forall x,y \in X_i\times Y_j, \underline{c}_{ij} \leq c(x,y) $,
\end{block}

\begin{block}{}
\begin{equation*}
\begin{aligned}
\underline{\mathcal{T}} = & \underset{ \pi_{ij} \in \mathbb{R}^{n} \otimes \mathbb{R}^{m}  }{\text{min}}
& & \sum_{i,j} \underline{c}_{ij} \pi_{ij}   \\
& \text{subject to} 
& & \forall i, \;  \underline{\mu}_i \alert{\leq} \sum_{j} \pi_{ij} \alert{\leq} \overline{\mu}_i , \\ 
& & & \forall j, \; \underline{\nu}_j  \alert{\leq} \sum_{i} \pi_{ij} \alert{\leq}  \overline{\nu}_j , \\
& & & \forall i, \forall j, \; \pi_{ij} \geq 0.
\end{aligned}
\end{equation*}
\end{block}


\begin{block}{}
 \[ \textrm{ then } \;\;\;\;\;\;\; \underline{\mathcal{T}} \leq \mathcal{T}_c(\mu,\nu). \]
\end{block}
\end{small}
}


\frame{
\frametitle{Example}
\begin{block}{}
$X=Y=[0,1]$\\
\begin{tabular}{cc}
 \includegraphics[width = 140 px]{image/true_example2.pdf}  &  \includegraphics[width = 140 px]{image/true_example.pdf}    \\
$\mu =1 \mathrm{d} x$ & $\nu = 1.5(1-y^2)  \mathrm{d} y$\\
\end{tabular}
\[ c(x,y) = \| x-y  \|^2  \]
\end{block}
}


\frame{
\begin{block}{}
\begin{figure}
\begin{center}
 \includegraphics[width = 190 px]{image/primal.pdf}
 \caption{Guaranteed lower bounds of $\mathcal{T}_c(\mu,\nu)$ where $n
   = \Card \{X_i\}_i $}
\end{center}
\end{figure}
\end{block}
}


\frame{ \frametitle{Outline} \tableofcontents }

\section{An  upper bound of the optimal value }

\subsection{Duality}


\frame{
\frametitle{Linear programming - Duality}
\begin{block}{Primal problem}
\begin{equation}
\begin{aligned}
& \underset{x \in \mathbb{R}^n }{\text{min}}
& & c^T x   \\
& \text{subject to} 
& &  Ax = b, \\
& &  &x \geq 0. \\
\end{aligned}
\end{equation}
\end{block}


\begin{block}{Dual problem}
\begin{equation}
\begin{aligned}
& \underset{y \in \mathbb{R}^m }{\text{max}}
& & b^T y  \\
& \text{subject to} & &  y_i \in \mathbb{R}, \\
& & & A^Ty  \leq c. \\
\end{aligned}
\end{equation}
\end{block}
}



\frame{
\frametitle{Duality}
\begin{small}
\begin{block}{}
\begin{equation*}
\begin{aligned}
 & \underset{\pi \in \mathcal{B} ( X \times Y )}{\text{inf}}
& & \int_{X \times Y} c(x,y) \mathrm{d} \pi(x,y)  \\
& \text{subject to} 
& &  \pi_X = \mu, 
\\ & & &\pi_Y = \nu \\
\end{aligned}
\label{transport}
\end{equation*}
\end{block}

\begin{block}{}
\begin{equation}
\begin{aligned}
 & \underset{\phi, \psi \in \mathcal{C}_b(X,Y ) }{\text{sup}}
& & \int_{X} \varphi (x) \, \mathrm{d} \mu (x) + \int_{Y} \psi (y) \, \mathrm{d} \nu (y)  \\
& \text{subject to} 
& &  \varphi (x) + \psi (y) \leq c(x, y). \\
\end{aligned}
\label{transport_dual}
\end{equation}
\end{block}

\begin{block}{}<2->
  where $\mathcal{C}_b(X,Y)$ denotes the set of all pairs of bounded and
  continuous functions $\phi : X \to \mathbb{R}$ and $\psi : Y \to
  \mathbb{R}$.
\end{block}

\begin{block}{}<3->
  If $X$ is compact and Haussdorff, $\mathcal{C}_b(X)^* = \{
  \textrm{Radon measure}  \}$
\end{block}
\end{small}

}


\frame{
\begin{block}{Kantorovich Duality}
  The minimum of the Kantorovich problem is equal to
\begin{equation}
\begin{aligned}
\mathcal{T}_c(\mu,\nu) = & \underset{\phi, \psi \in \mathcal{C}_b(X,Y ) }{\text{sup}}
& & \int_{X} \varphi (x) \, \mathrm{d} \mu (x) + \int_{Y} \psi (y) \, \mathrm{d} \nu (y)  \\
& \text{subject to} 
& &  \varphi (x) + \psi (y) \leq c(x, y). \\
\end{aligned}
\label{transport_dual}
\end{equation}
\end{block}
}

\frame{
\begin{block}{Interpretation in the discrete case}
\begin{equation}
\begin{aligned}
& \underset{ (\phi_i) \in \mathbb{R}^n, (\psi_j) \in \mathbb{R}^m  }{\text{sup}}
& & \sum_{i} \phi_i {\mu}_i + \sum_{j} \psi_j  {\nu}_i  \\
& \text{subject to} 
& &  \phi_i + \psi_j \leq c_{ij}\\
\end{aligned}
\label{transport_dual}
\end{equation}
\begin{center}
 \includegraphics[width = 160 px]{image/dual_books_2.pdf}
\end{center}
\end{block}
}

\frame{
\begin{block}{Interpretation in the discrete case}
\begin{equation}
\begin{aligned}
& \underset{ (\phi_i) \in \mathbb{R}^n, (\psi_j) \in \mathbb{R}^m  }{\text{sup}}
& & \sum_{i} \phi_i {\mu}_i + \sum_{j} \psi_j  {\nu}_i  \\
& \text{subject to} 
& &  \phi_i + \psi_j \leq c_{ij}\\
\end{aligned}
\label{transport_dual}
\end{equation}
\begin{center}
 \includegraphics[width = 160 px]{image/dual_books_3.pdf}
\end{center}
\end{block}
}

\frame{
\begin{block}{Interpretation in the discrete case}
\begin{equation}
\begin{aligned}
& \underset{ (\phi_i) \in \mathbb{R}^n, (\psi_j) \in \mathbb{R}^m  }{\text{sup}}
& & \sum_{i} \phi_i {\mu}_i + \sum_{j} \psi_j  {\nu}_i  \\
& \text{subject to} 
& &  \phi_i + \psi_j \leq c_{ij}\\
\end{aligned}
\label{transport_dual}
\end{equation}
\begin{center}
 \includegraphics[width = 160 px]{image/dual_books_4.pdf}
\end{center}
\end{block}
}



\subsection{Finite dimensional relaxation}

\frame{
\begin{small}
\begin{block}{Proposition - Relaxation}
  Let $\mu$ and $\nu$ (with support $X$ and $Y$) be absolutely
  continuous measures with respect to Lebesgue measure. If $\{X_i\}_i$
  and $\{Y_i\}_i$ be finite pavings of $X$ and $Y$. Suppose that
  $\mu(X_i) \in [\underline{\mu}_i, \overline{\mu}_i]$, $\nu(Y_j) \in [\underline{\nu}_j, \overline{\nu}_j]$,
  and $\forall x,y \in X_i\times Y_j,  c(x,y) \leq \overline{c}_{ij}   $,
\end{block}

\begin{block}{}<2->
\begin{equation}
\begin{aligned}
\overline{\mathcal{T}} = & \underset{ (\phi_i) \in \mathbb{R}^n, (\psi_j) \in \mathbb{R}^m  }{\text{sup}}
& & \sum_{i} \phi_i \overline{\mu}_i + \sum_{j} \psi_j  \overline{\nu}_i  \\
& \text{subject to} 
& &  \phi_i + \psi_j \leq \overline{c}_{ij}\\
\end{aligned}
\label{transport_dual}
\end{equation}
\end{block}

\begin{block}{}<3->
 \[ \textrm{ then } \;\;\;\;\;\;\;  \mathcal{T}_c(\mu,\nu) \leq \overline{\mathcal{T}} . \]
\end{block}
\end{small}
}


\frame{
\begin{block}{}
\begin{figure}
\begin{center}
 \includegraphics[width = 300 px]{image/both.pdf}
 \caption{Guaranteed upper bounds of $\mathcal{T}_c(\mu,\nu)$ where $n
   = \Card \{X_i\}_i $}
\end{center}
\end{figure}
\end{block}
}




\section{Conclusion - Future work}
\frame{
\begin{block}{Software}
\begin{itemize}
\item filib - FI\_LIB - A fast interval library, \\
  \begin{tiny}\url{http://www2.math.uni-wuppertal.de/~xsc/software/filib.html}\end{tiny}
\item GLPK - GNU Linear Programming Kit (GLPK),\\
  \begin{tiny}\url{http://www.gnu.org/software/glpk/}\end{tiny}
\item GMP - GNU Multiple Precision Arithmetic Library,\\
  \begin{tiny}\url{https://gmplib.org/}\end{tiny}
\item Source code is available on my webpage.
\end{itemize}
\end{block}
}

\frame{
\begin{block}<1->{Future work}
\begin{itemize}
\item Compute guaranteed enclosures of the solution combining linear
  programming and constraints propagation.
\item <2-> Generalize this methodology to other problems (D. Henrion \& J.B. Lasserre): 
\begin{itemize}
\item <3-> Probability and Markov Chains
\item <4-> Optimal Control with occupation measures (ODE),
\item <5-> Others as in \emph{ Moments, Positive Polynomials and Their
    Applications}, J.B Lasserre, Imperial College Press Optimization
  Series (2009)
\end{itemize}
\end{itemize}
\end{block}


\begin{block}<6->{}
Merci pour votre attention.
\end{block}
}











\end{document}

