The approach detailed previously requires to compute 
the variable that minimizes four convex functions
on constraint domains, one  
for each primal variable $q_i$, $P_{sh}$, $R_h$, and $x_{hl}$,
Among state of the art for this optimization step, there is
the  L-BFGS-B algorithm~\cite{byrd1995limited}, 
the truncated Newton algorithm, 
the Constrained Optimization BY Linear Approximation (COBYLA) method~\cite{ANU:1770520}.
However, all these methods suffer from being iterative approaches 
each iteration including  many steps of computation to obtain an approximation 
of the minimal value. 
This approach is dramatic since the objective is to 
reduce all the computation steps to increase the network lifetime. 
  
A closer look to each function that has to be minimized shows that it is
differentiable and the minimal value can be computed in only one step. 
The table~\ref{table:min} presents these minimal value for each primal
variable.
Thanks to this formal calculus, computing the 
new iterate of each primal variable  only requires 
one computation step.

\begin{table*}[t]
$$
\begin{array}{|l|l|l|}
\hline
q_i^{(k+1)} &
 \arg\min_{q>0}
\left(
q^2 + q. 
\left(
\sum_{l \in L } a_{il}w_l^{(k)}-
\lambda_i^{(k)}B_i
\right)
\right)
 & 
\max \left\{\epsilon,\dfrac{\sum_{l \in L } a_{il}w_l^{(k)}-
\lambda_i^{(k)}B_i}{2}\right\} \\
\hline
P_{sh}^{(k+1)}& 
\arg \min_{p > 0} 
\left(
v_h^{(k)}.\dfrac{\ln(\sigma^2/D_h)}{\gamma p^{2/3}} + \lambda_h^{(k)}p
+ \delta_p p^{8/3}
\right)
&
\max \left\{\epsilon,
\left(
\dfrac{
-3\lambda_h^{(k)} + \sqrt{(3\lambda_h^{(k)})^2 + 64\delta_p/\gamma.\ln(\sigma^2/D_h)}
}{16\delta_p}
\right)^{\frac{3}{5}}
\right\} \\
\hline
R_h^{(k+1)}
&
\arg \min_{r \geq 0 }
\left(
\delta_r r^2 
-v_h^{(k)}.r - \sum_{i \in N} u_{hi}^{(k)} \eta_{hi}
\right) &
\max\left\{0,\dfrac{v_h^{(k)}}{2\delta_r}\right\}
\\
\hline
x_{hl}^{(k+1)} &
\begin{array}{l}
\arg \min_{x \geq 0}
\left(
\delta_x.x^2  \right.\\
\qquad \qquad + x.
\sum_{i \in N} \left( 
\lambda_{i}^{(k)}.(c^s_l.a_{il}^{+} +
c^r. a_{il}^{-} ) \right.\\
\qquad \qquad\qquad \qquad +
\left.\left. u_{hi}^{(k)} a_{il}
\right)
\right)
\end{array}
&
\max\left\{0,\dfrac{-\sum_{i \in N} \left( 
\lambda_{i}^{(k)}.(c^s_l.a_{il}^{+} +
c^r. a_{il}^{-} ) + u_{hi}^{(k)} a_{il}
\right)}{2\delta_x}\right\}
\\
\hline
\end{array}
$$
\caption{Primal Variables: Argmin and Direct Calculus}\label{table:min}
\end{table*}


This improvment has been evaluated on a set of experiments.
For 10 tresholds $t$, such that $1E-5 \le t \le 1E-3$, we have 
executed 10 times the aproach detailled before either with the new 
gradient calculus or with the original argmin one. 
The Table~\ref{Table:argmin:time} summarizes the averages of these 
excution times, given in seconds. We remark time spent with the gradient 
approach is about 37 times smaller than the one of the argmin one.
Among implementations of argmin aproaches, we have retained 
the COBYLA one since it does not require any gradient to be executed.

\begin{table*}
\begin{scriptsize}
$$
\begin{array}{|l|l|l|l|l|l|l|l|l|l|l|}
\hline
\textrm{Convergence Treshold} &
10^{-5} &
1.67.10^{-5} &
2.78.10^{-5} &
4.64.10^{-5} &
7.74.10^{-5} &
1.29.10^{-4} &
2.15.10^{-4} &
3.59.10^{-4} &
5.99.10^{-4} &
0.001 \\
\hline 
\textrm{Gradient Calculus} &
56.29 &
29.17 &
37.05 &
6.05 &
5.47 &
3.82 &
1.91 &
2.37 &
0.87 &
0.65 \\
\hline
\textrm{Argmin Method} &  
2224.27 &
1158.37 &
1125.21&
216.82&
162.26&
126.99&
58.044&
74.204&
23.99&
15.85\\
\hline
\end{array}
$$
\caption{Convergence Times for Gradient and Argmin Methods}\label{Table:argmin:time}
\end{scriptsize}
\end{table*}

 