beaucoup decourbes

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

diff --git a/IWCMC14/convergence.tex b/IWCMC14/convergence.tex
index 13b3b2d6a14c0a7394fd2df507baba0a2a0c472e..b689f9900f32e5deec97eb399cd7eadd7a0e75d4 100644 (file)
--- a/IWCMC14/convergence.tex
+++ b/IWCMC14/convergence.tex
@@ -1,32 +1,42 @@
 Let us first have a discussion on the stop criterion of the cited algorithm.
 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 is less than a given 
+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)
 may indeed easily fall into a local trap because some of introduced 
 variables may lead to uniformity of the output.
 
 \begin{figure}
 threshold, this does not ensure that the lifetime has been maximized.
 Minimizing a function on a multiple domain (as the dual function)
 may indeed easily fall into a local trap because some of introduced 
 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}  
 

-Experiments have indeed shown that even if the dual
-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 
-$\frac{\max_{i \in N} q_i} {\min_{i \in N}q_i}-1$--
+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 
 is represented in $y$-coordinate 

- with respect to the
+with respect to the

 value of the threshold for dual function that is represented in 
 $x$-coordinate.
 value of the threshold for dual function that is represented in 
 $x$-coordinate.

-This figure shows that a very small threshold is a necessary condition, but not 
-a sufficient criteria to observe convergence of $q_i$.

 
 

+
+Experiments shown that even if the dual
+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, \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 $q_i$.

 In the following, we consider the system are $\epsilon$-stable  if both 
 maximum amplitude rate and the dual function are less than a threshold 
 $\epsilon$.
 In the following, we consider the system are $\epsilon$-stable  if both 
 maximum amplitude rate and the dual function are less than a threshold 
 $\epsilon$.