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.
Minimizing a function on a multiple domain (as the dual function)
may indeed easilly 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}
  \label{fig:convergence:scatterplot}
\end{figure}  

Experimentations 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 
$\frac{\max_{i \in N} q_i} {\min_{i \in N}q_i}-1$--
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 
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 threeshold 
$\epsilon$.


  