+The approach detailed previously requires to compute
+the variable that minimizes four convex fonctions
+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*}
+
+
\ No newline at end of file
