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 
and need many steps of computation to obtain an approximation 
of the minimal value. This approach is dramatic whilst the objective is to 
reduce all the computation steps. 
  
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.

\begin{table*}[t]
$$
\begin{array}{|l|l|l|}
\hline
q_i^{(k)} &
 \arg\min_{q_i>0}
\left(
q^2 + q. 
\left(
\sum_{l \in L } a_{il}w_l^{(k)}-
\lambda_i^{(k)}B_i
\right)
\right)
 & 
\max \left(\epsilon,\dfrac{\sum_{l \in L } a_{il}w_l^{(k)}-
\lambda_i^{(k)}B_i}{2}\right) \\
\hline
P_{sh}^{(k)}& 
\arg \min_{p > 0} 
\left(
v_h^{(k)}.\dfrac{\ln(\sigma^2/D_h)}{\gamma p^{2/3}} + \lambda_h^{(k)}p
+ \delta_p p^{8/3}
\right)
&
\max \left(\epsilon,
\left(
\dfrac{
-\lambda_h^{(k)} + \sqrt{(\lambda_h^{(k)})^2 + \dfrac{64}{9}\alpha}
}{\frac{16}{3}\delta_p}
\right)^{\frac{3}{5}}
\right) \\
\hline
R_h^{(k)}
&
\arg \min_{r \geq 0 }
\left(
\delta_r r^2 
-v_h^{(k)}.r - \sum_{i \in N} u_{hi}^{(k)} \eta_{hi}
\right) &
\max\left(0,\dfrac{v_h^{(k)}}{2\delta_r}\right)
\\
\hline
x_{hl}^{(k)} &
\begin{array}{l}
\arg \min_{x \geq 0}
\left(
\delta_x.x^2  \right.\\
\qquad \qquad + x.
\sum_{i \in N} \left( 
\lambda_{i}^{(k)}.(c^s_l.a_{il}^{+} +
c^r. a_{il}^{-} ) \right.\\
\qquad \qquad\qquad \qquad +
\left.\left. u_{hi}^{(k)} a_{il}
\right)
\right)
\end{array}
&
\max\left(0,\dfrac{-\sum_{i \in N} \left( 
\lambda_{i}^{(k)}.(c^s_l.a_{il}^{+} +
c^r. a_{il}^{-} ) + u_{hi}^{(k)} a_{il}
\right)}{2\delta_x}\right)
\\
\hline
\end{array}
$$
\caption{Expression of each optimized primal variable}
\end{table*}

  