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. 

  