In the algorithm presented in the previous section,  
the encoding power consumption is iteratively updated with 
$
P_{sh}^{(k)} 
=
\arg \min_{p > 0} 
\left(
v_h^{(k)}.\dfrac{\ln(\sigma^2/D_h)}{\gamma p ^{2/3}} + \lambda_h^{(k)}p
\right)
$.
The function inside the $\arg \min$ is strictly convex if and only if 
$\lambda_h$ is not null. This asymptotic configuration may arise due to 
the definition of $\lambda_h$. Worth, in this case,  the function is 
strictly decreasing and the minimal value is obtained when $p$ is the infinity.

To prevent this configuration, we replace the objective function given 
in equation~(\ref{eq:obj2}) by 
\begin{equation}
\sum_{i \in N }q_i^2 + 
\delta_x \sum_{h \in V, l \in L } x_{hl}^2 
+ \delta_r\sum_{h \in V }R_{h}^2
+ \delta_p\sum_{h \in V }P_{sh}^{\frac{8}{3}}.
\label{eq:obj2p}
\end{equation}
In this equation we have first introduced new regularisation factors
(namely $\delta_x$, $\delta_r$, and $\delta_p$)
instead of the sole $\delta$.  
This allows to  further separately study the influence of each factor.
Next, the introduction of the rational exponent is motivated by the goal of 
providing a strictly convex function.

Let us now verify that the induced function is convex.
Let $f: \R^{+*} \rightarrow \R$ such that $
f(p)= v_h.\dfrac{\ln(\sigma^2/D_h)}{\gamma p^{2/3}} + \lambda_h p
+ \delta_p p^{8/3}$. This function is differentiable and 
for any $x \in \R^{+*}$ and we have
$$
\begin{array}{rcl}
f'(p) &=& -2/3.v_h.\dfrac{\ln(\sigma^2/D_h)}{\gamma p^{5/3}} + \lambda_h + 
8/3.\delta_p p^{5/3} \\
& = & \dfrac {8/3.\delta_p p^{10/3} + \lambda_h p^{5/3} -2/3\gamma.v_h\ln(\sigma^2/D_h)  }{p^{5/3}}
\end{array}
$$
which is positive if and only if the numerator is.
Provided $p^{5/3}$ is replaced by $P$, we have a quadratic function 
which is strictly convex, for any value of $\lambda_h$ since the discriminant 
is positive. 

This proposed enhacement has been evaluated as follows:  
10 tresholds $t$, such that $1E-5 \le t \le 1E-3$, have 
been selected and for each of them,  
10 random configurations have been generated.
For each one, we store the 
number of iterations which is sufficient to make the dual 
function variation smaller than this given treshold with 
the two approaches: either the original one ore the
one which is convex garantee.

The Figure~\ref{Fig:convex} summarizes the average number of convergence 
iterations for each tresholdvalue. As we can see, even if this new 
enhanced method introduces new calculus, 
it only slows few down the algorithm and garantee the convexity, 
and thus the convergence.

\begin{figure*}
\begin{center}
\includegraphics[scale=0.5]{convex.png}
\end{center}
\caption{Original Vs Convex Garantee Approaches}\label{Fig:convex}
\end{figure*} 


