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 stricly convex if and only if 
$\lamda_h$ is not null. This asymptotic configuration may arrise due to 
the definition of $\lambda_i$. Worth, in this case,  the function is 
stricly 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 }\delta.R_{h}^2
+ \delta_p\sum_{h \in V }\delta.P_{sh}^{\frac{8}{3}}.
\label{eq:obj2}
\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 study the influence of each modification separately.
Next, the introduction of the rationnal exponent is motivated by the goal of 
providing a stricly convex function.

  