modification de refs

[desynchronisation-controle.git] / IWCMC14 / convergence.tex

diff --git a/IWCMC14/convergence.tex b/IWCMC14/convergence.tex
index ea4d4c223a2e597bf7f319f08c9dd60b8cfb30b2..b689f9900f32e5deec97eb399cd7eadd7a0e75d4 100644 (file)
--- a/IWCMC14/convergence.tex
+++ b/IWCMC14/convergence.tex
@@ -1,34 +1,44 @@
-Let us first have a discussion on the stop criterion of the citted algorithm.
-We claim that even if the variation of the dual function is less than a given 
-threeshold, this does not ensure that the lifetime has been maximized.
+Let us first have a discussion on the stop criterion of the cited algorithm.
+We claim that even if the variation of the dual function
+(recalled in equation (\ref{eq:dualFunction}))
+is less than a given 
+threshold, this does not ensure that the lifetime has been maximized.

 Minimizing a function on a multiple domain (as the dual function)
 Minimizing a function on a multiple domain (as the dual function)

-may indeed easilly fall into a local trap because some of introduced 
+may indeed easily fall into a local trap because some of introduced 

 variables may lead to uniformity of the output.
 
 \begin{figure}
 variables may lead to uniformity of the output.
 
 \begin{figure}

-  to be continued 
-  \caption{Relations between dual function threshold and $q_i$ convergence}
+  \begin{center}
+    \includegraphics[scale=0.5]{amplrate.png}
+  \end{center}
+  \caption{Relations between dual function variation and convergence of all the $q_i$}

   \label{fig:convergence:scatterplot}
 \end{figure}  
 
   \label{fig:convergence:scatterplot}
 \end{figure}  
 

-Experimentations have indeed shown that even if the dual
+To explain this, we introduce the maximum amplitude rate $\zeta$ 
+of the sequence of $q$ which is defined as  
+$$
+\dfrac{\max_{i \in N} \{q_i\}}
+{\min_{i \in N} \{q_i\}}-1.
+$$
+The Figure~\ref{fig:convergence:scatterplot} presents 
+a scatter plot between $\zeta$, which 
+is represented in $y$-coordinate 
+with respect to the
+value of the threshold for dual function that is represented in 
+$x$-coordinate.
+
+
+Experiments shown that even if the dual

 function seems to be constant 
 (variations between two evaluations of this one is less than $10^{-5}$) 
 function seems to be constant 
 (variations between two evaluations of this one is less than $10^{-5}$) 

-not all the $q_i$ have the same value.
-For instance, the Figure~\ref{fig:convergence:scatterplot} presents 
-a scatter plot.
-
-The maximum amplitude rate  of the sequence of $q_i$ --which is 
-$\frac{\max_{i \in N} q_i} {\min_{i \in N}q_i}-1$--
-is represented in $y$-coordonates 
- with respect to the
-value of the threeshold for dual function that is represented in 
-$x$-coordonates.
-This figure shows that a very small threshold is a necessary condition, but not 
+not all the $q_i$ have the same value, \textit{i. e.}, $\zeta$ is still large.
+For instance, even with a threshold set to $10^{-5}$ there still can be more than
+45\% of differences between two $q_i$. 
+To summarize, a very small threshold is a necessary condition, but not 

 a sufficient criteria to observe convergence of $q_i$.
 a sufficient criteria to observe convergence of $q_i$.

-

 In the following, we consider the system are $\epsilon$-stable  if both 
 In the following, we consider the system are $\epsilon$-stable  if both 

-maximum amplitude rate and the dual function are less than a threeshold 
+maximum amplitude rate and the dual function are less than a threshold 

 $\epsilon$.
 
 
 $\epsilon$.