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}
- 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}
-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
- with respect to the
+with respect to the
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
$\epsilon$.
