Trouver $R$, $x$, $P_s$, $q_i$  minimisant 
$$
\begin{array}{l}
  L(R,x,P_s,q,u,v,\lambda,w)= \\
 \sum_{i \in N }q_i^2 + \sum_{h \in V, l \in L }\delta.x_{hl}^2 
+ \sum_{h \in V }\delta.R_{h}^2  \\
+ \sum_{h \in V }\sum_{i \in N } u_{hi} \left(\sum_{l \in L }a_{il}x_{hl} - \eta_{hi}\right) \\
+ \sum_{h \in V}v_{h}.\left( \dfrac{\ln(\sigma^2/D_h)}{\gamma.P_{sh}^{2/3}} - R_h  \right) \\
+ \sum_{i \in N} \lambda_{i}. \left( P_{si}+ \sum_{l \in L}a_{il}^{+}.c^s_l.\left( \sum_{h \in V}x_{hl} \right)  \right.
\\
\qquad \qquad \qquad + \left. \sum_{l \in L} a_{il}^{-}.c^r.\left( \sum_{h \in V}x_{hl} \right) - q.B_i \right) \\
+ \sum_{l \in L} w_l. \left( \sum_{i \in N} a_{il}.q_i \right)
\end{array}
$$
\begin{itemize}
\item $x_{hl}\geq0, \forall h \in V, \forall l \in L$

\item $R_h \geq 0, \forall h \in V$

\item $P_{sh} > 0,\forall h \in V$

\item $q_i > 0,\forall i \in N$

\end{itemize}