\item $P_{sh} > 0,\forall h \in V$
\end{enumerate}
-
To achieve this optimizing goal
a local optimisation, the problem is translated into an
equivalent one: find $R$, $x$, $P_s$ which minimize
\end{array}
$$
-The authors then apply a dual based approach with Lagrange multiplier
-to solve such a problem.
-They first introduce dual variables
-$u_{hi}$, $v_{h}$, $\lambda_{i}$, and $w_l$ for any
-$h \in V$,$ i \in N$, and $l \in L$.
-They next replace the objective of reducing $\sum_{i \in N }q_i^2$
+
+They thus replace the objective of reducing
+$\sum_{i \in N }q_i^2$
by the objective of reducing
-$$
+\begin{equation}
\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
-$$
-where $\delta$ is a \JFC{ formalisation} factor.
-This allows indeed to get convex functions whose minimum value is unique.
+\label{eq:obj2}
+\end{equation}
+where $\delta$ is a regularisation factor.
+This indeed introduces quadratic fonctions on variables $x_{hl}$ and
+$R_{h}$ and makes some of the functions strictly convex.
+
+The authors then apply a classical dual based approach with Lagrange multiplier
+to solve such a problem~\cite{}.
+They first introduce dual variables
+$u_{hi}$, $v_{h}$, $\lambda_{i}$, and $w_l$ for any
+$h \in V$,$ i \in N$, and $l \in L$.
-The proposed algorithm iteratively computes the following variables
+\begin{equation}
+\begin{array}{l}
+L(R,x,P_{s},q,u,v,\lambda,w)=\\
+\sum_{i \in N} q_i^2 + q_i. \left(
+\sum_{l \in L } a_{il}w_l^{(k)}-
+\lambda_iB_i
+\right)\\
++ \sum_{h \in V}
+v_h.\dfrac{\ln(\sigma^2/D_h)}{\gamma P_{sh} ^{2/3}} + \lambda_h P_{sh}\\
++ \sum_{h \in V} \sum_{l\in L}
+\left(
+\delta.x_{hl}^2 \right.\\
+\qquad \qquad + x_{hl}.
+\sum_{i \in N} \left(
+\lambda_{i}.(c^s_l.a_{il}^{+} +
+c^r. a_{il}^{-} ) \right.\\
+\qquad \qquad\qquad \qquad +
+\left.\left. u_{hi} a_{il}
+\right)
+\right)\\
+ + \sum_{h \in V}
+\delta R_{h}^2
+-v_h.R_{h} - \sum_{i \in N} u_{hi}\eta_{hi}
+\end{array}
+\end{equation}
+The proposed algorithm iteratively computes the following variables
+untill the variation of the dual function is less than a given threshold.
\begin{enumerate}
\item $ u_{hi}^{(k+1)} = u_{hi}^{(k)} - \theta^{(k)}. \left(
\eta_{hi}^{(k)} - \sum_{l \in L }a_{il}x_{hl}^{(k)} \right) $
$v_{h}^{(k+1)}= \max\left\{0,v_{h}^{(k)} - \theta^{(k)}.\left( R_h^{(k)} - \dfrac{\ln(\sigma^2/D_h)}{\gamma.(P_{sh}^{(k)})^{2/3}} \right)\right\}$
\item
$\begin{array}{l}
- \lambda_{i}^{(k+1)} = \lambda_{i}^{(k)} - \theta^{(k)}.\left(
- q^{(k)}.B_i \right.\\
+ \lambda_{i}^{(k+1)} = \max\left\{0, \lambda_{i}^{(k)} - \theta^{(k)}.\left(
+ q^{(k)}.B_i \right. \left.\\
\qquad\qquad\qquad -\sum_{l \in L}a_{il}^{+}.c^s_l.\left( \sum_{h \in V}x_{hl}^{(k)} \right) \\
- \qquad\qquad\qquad - \left. \sum_{l \in L} a_{il}^{-}.c^r.\left( \sum_{h \in V}x_{hl}^{(k)} \right) - P_{si}^{(k)} \right)
+ \qquad\qquad\qquad - \left.\left. \sum_{l \in L} a_{il}^{-}.c^r.\left( \sum_{h \in V}x_{hl}^{(k)} \right) - P_{si}^{(k)} \right) \right\}
\end{array}
$
\item
$q_i^{(k)} = \arg\min_{q_i>0}
\left(
-q^2 + q.
+q_i^2 + q_i.
\left(
\sum_{l \in L } a_{il}w_l^{(k)}-
\lambda_i^{(k)}B_i
\right)
\right)$
-\item
+\item \label{item:psh}
$
P_{sh}^{(k)}
=
$
\end{enumerate}
where the first four elements are dual variable and the last four ones are
-primal ones
\ No newline at end of file
+primal ones.
\ No newline at end of file
