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)
may indeed easily fall into a local trap because some of introduced 
variables may lead to uniformity of the output.

\begin{figure}
  \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}  

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 $10^{-5}$) 
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 
$\epsilon$.


  