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Aliaume Lopez
projet-optimisation
Commits
6120484e
Commit
6120484e
authored
Jan 10, 2017
by
Gaetan D
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c425bb7d
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6120484e
...
...
@@ -364,15 +364,6 @@
\end{exampleblock}
\end{frame}
\subsection
{
Ajout de contraintes pour l'intégralité
}
\begin{frame}
\frametitle
{
Contraintes supplémentaires
}
\begin{block}
{
coucou
}
Copier ici ce qu'on lira de la présentation de Nathanaël
\end{block}
\end{frame}
\begin{frame}
...
...
@@ -441,9 +432,25 @@
\begin{frame}
\frametitle
{
Simplexe
}
Implémentation du simplexe maison
Les performances dégeu (quantifier !)
\begin{block}
{
Implémentation
}
\begin{itemize}
\item
PL sous forme standard : Maximiser
$
c
^
Tx
$
avec
$
Ax
=
b
$
,
$
x
\ge
0
$
\item
règle de Bland pour le pivotage
\item
résoudre un problème de Phase 1, et un autre en Phase 2
\end{itemize}
\end{block}
\ds
Mais des performances immondes.
\end{frame}
\begin{frame}
...
...
@@ -467,23 +474,33 @@
\begin{frame}
\frametitle
{
Coupe minimale
}
In the MinimumCutPhase, the subset A of the graphs vertices grows starting with an arbitrary single vertex until
A is equal to V. In each step, the vertex which is outside of A, but most tightly connected with A
is added to the set A.
Shrink G by merging the two vertices added last.
After one phase of the MinimumCutPhase, the two vertices are merged as a new
vertex, and edges from the two vertices to a remaining vertex are replaced by
an edge weighted by the sum of the weights of the previous two edges. Edges
joining the merged nodes are removed. If there is a minimum cut of G
separating s and t, the C is a minimum cut of G. If not, then the minimum cut
of G must have s and t on a same side. Therefore, the algorithm would merge
them as one node.
\[
O
(
n
(
m
+
n
)
\log
n
)
\]
\begin{block}
{
Implémentation "naïve"
}
Recherche des coupes minimales entre tous les couples de sommets.
O(
$
|V|
^
3
|E|
^
2
$
) (avec Edmonds-Karp)
\end{block}
\ds
Des performances suffisantes pour un "mauvais" simplexe.
\es
\begin{block}
{
Stoer-Wagner
}
Tout faire en même temps.
\ds
Complexité : O(
$
|V|
(
|E|
+
|V|
\log
|V|
)
$
).
\end{block}
\ds
Implémenté dans NetworkX.
\end{frame}
\section
{
Résultats obtenus
}
...
...
@@ -516,17 +533,13 @@
\section
{
Dualité
}
\subsection
{
Calcul du problème dual
}
\begin{frame}
\frametitle
{
Expression du problème dual
}
\begin{equation}
\label
{
eqn:dual
}
\boxed
{
\begin{array}
{
rll
}
\textrm
{
max
}
&
\displaystyle
-
\sum
_{
uv
\in
E'
}
\lambda
_{
1,uv
}
x
_{
uv
}
- 2
\sum
_{
u
\in
V
}
\nu
_
u + 2
\sum
_{
S
\in
W
}
\lambda
_{
3,S
}
\\
&
\lambda
_
1
\geq
0
\\
\textrm
{
Maximiser
}
&
\displaystyle
-
\sum
_{
uv
\in
E'
}
\lambda
_{
1,uv
}
x
_{
uv
}
- 2
\sum
_{
u
\in
V
}
\nu
_
u + 2
\sum
_{
S
\in
W
}
\lambda
_{
3,S
}
\\
\textrm
{
Avec
}
&
\lambda
_
1
\geq
0
\\
&
\lambda
_
2
\geq
0
\\
&
\lambda
_
3
\geq
0
\\
&
c
_{
E'
}
-
\lambda
_
1 +
\lambda
_
2 +
{}^
t D
_{
E'
}
\nu
-
{}^
t G
_{
E'
}
\lambda
_
3 = 0
...
...
@@ -555,6 +568,14 @@ Avec $D_{E'}$ la matirce d'incidence du graphe restreint à $E'$, et $G_{E'}$ la
\end{frame}
\section*
{
Conclusion
}
\begin{frame}
\frametitle
{
Conclusion
}
\begin{block}
{
Contraintes supplémentaires
}
Copier ici ce qu'on lira de la présentation de Nathanaël
\end{block}
\end{frame}
\section*
{
Bibliographie
}
...
...
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