Let us first the basic recalls of the~\cite{HLG09} article.


The  precise the context of video sensor network as represented for instance 
in figure~\ref{fig:sn}.

\begin{figure}
\begin{center}
\includegraphics[scale=0.5]{reseau.png}
\caption{SN with 10 sensor}\label{fig:sn}.
\end{center}
\end{figure} 


Let us give a formalisation of such a video network sensor.
We start with the flow formalising:

The video sensor network is represented as a strongly 
connected oriented labelled graph. 
In this one, 
the nodes, in a set $N$ are sensors, links, or the sink.
Furthermore, there is an edge from $i$ to $j$ if $i$ can 
send a message to $j$. The set of all edges is further denoted as
$L$ .  
This boolean information is stored as a  
matrix $A=(a_{il})_{i \in N, l \in L}$,
where 
$a_{il} = 
\left\{
    \begin{array}{rl}
      1 & \textrm{if $l$ starts with $i$ } \\
      -1 & \textrm{si $l$ ends width $i$ }  \\
      0 & \textrm{otherwise}
    \end{array}
  \right.$.


Let $V \subset N $ be the set of the video sensors of $N$.
Let thus $R_h$, $R_h \geq 0$  be the encoding rate of  video sensor $h$, $h \in V$.  
Let $\eta_{hi}$ be the  production rate of the $i$ node, for the $h$ session. More precisely, we have 
  $$
\eta_{hi} = 
\left\{
    \begin{array}{rl}
      R_h & \textrm{if $i$ is $h$} \\
      -R_h & \textrm{if $i$ is the sink} \\
      0 & \textrm{otherwise}
    \end{array}
  \right.$$
  
We are then left to focus on the flows in this network.
Let $x_{hl}$, $x_{hl}\geq 0$, be the flow inside the edge $l$ that 
issued from the $h$ session and 
let $y_l = \sum_{h \in V}x_{hl} $ the sum of all the flows inside $l$.
Thus, what is produced inside the $i^{th}$ sensor for session $h$ 
is  $ \eta_{hi} = \sum_{l \in L }a_{il}x_{hl} $.


The encoding power of the $i$ node is $P_{si}$, $P_{si} > 0$.

The distortion is bounded $\sigma^2 e^{-\gamma . R_h.P_{sh}^{}2/3} \leq D_h$.

The initial energy of the $i$ node is  $B_i$.

The overall consumed power of the $i$ node is 
$P_{si}+ P_{ti} + P_{ri}= 
P_{si}+ \sum_{l \in L}a_{il}^{+}.c^s_l.y_l + 
\sum_{l \in L} a_{il}^{-}.c^r.y_l \leq q.B_i. 
$

The objective is thus to find $R$, $x$, $P_s$  which minimize
 $q$ under the following set of constraints
\begin{enumerate}
\item $\sum_{l \in L }a_{il}x_{hl} = \eta_{hi},\forall h \in V, \forall i \in N  $
\item $ \sum_{h \in V}x_{hl} = y_l,\forall l \in L$
\item $\dfrac{\ln(\sigma^2/D_h)}{\gamma.P_{sh}^{2/3}} \leq R_h \forall h \in V$
\item \label{itm:q} $P_{si}+ \sum_{l \in L}a_{il}^{+}.c^s_l.y_l + 
\sum_{l \in L} a_{il}^{-}.c^r.y_l \leq q.B_i, \forall i \in N$
\item $x_{hl}\geq0, \forall h \in V, \forall l \in L$
\item $R_h \geq 0, \forall h \in V$
\item $P_{sh} > 0,\forall h \in V$
\end{enumerate}


To achieve a local optimisation, the problem is translated into an 
equivalent one: find $R$, $x$, $P_s$  which minimize 
$\sum_{i \in N }q_i^2$ with the same set of constraints, but  
item \ref{itm:q}, which is replaced by:

$$P_{si}+ \sum_{l \in L}a_{il}^{+}.c^s_l.\left( \sum_{h \in V}x_{hl} \right) + 
\sum_{l \in L} a_{il}^{-}.c^r.\left( \sum_{h \in V}x_{hl} \right) \leq q.B_i, \forall i \in N$$


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 $\sum_{i \in N }q_i^2$
by the objective of reducing 
$$
\sum_{i \in N }q_i^2 + 
\sum_{h \in V, l \in L } \delta.x_{hl}^2 
+ \sum_{h \in V }\delta.R_{h}^2 
$$
where $\delta$ is a \JFC{ formalisation} factor.
This allows indeed to get convex functions whose minimum value is unique.

The proposed algorithm iteratively computes the following variables 

\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) $
\item 
$v_{h}^{(k+1)}= \max\left\{0,v_{h}^{(k)} -  \theta^{(k)}.\left( R_h^{(k)} - \dfrac{\ln(\sigma^2/D_h)}{\gamma.(P_{sh}^{(k)})^{2/3}}   \right)\right\}$
\item 
  $\begin{array}{rcl}
   \lambda_{i}^{(k+1)} = \lambda_{i}^{(k)} - \theta^{(k)}&.&\left( 
    q^{(k)}.B_i - 
    \sum_{l \in L}a_{il}^{+}.c^s_l.\left( \sum_{h \in V}x_{hl}^{(k)} \right) \right.  \\
   && - \left. \sum_{l \in L} a_{il}^{-}.c^r.\left( \sum_{h \in V}x_{hl}^{(k)} \right) - P_{si}^{(k)}  \right)
\end{array}
$

\item 
$w_l^{(k+1)} = w_l^{(k+1)} +  \theta^{(k)}. \left( \sum_{i \in N} a_{il}.q_i^{(k)} \right)$


\item 
$\theta^{(k)} = \omega / t^{1/2}$

 \item 
$q_i^{(k)} = \arg\min_{q_i>0}
\left(
q^2 + q. 
\left(
\sum_{l \in L } a_{il}w_l^{(k)}-
\lambda_i^{(k)}B_i
\right)
\right)$

\item 
$
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
\right)
$

\item 
$
R_h^{(k)}
=
\arg \min_{r \geq 0 }
\left(
\delta r^2 
-v_h^{(k)}.r - \sum_{i \in N} u_{hi}^{(k)} \eta_{hi}
\right)
$
\item 
$
x_{hl}^{(k)} =
\arg \min_{x \geq 0}
\left(
\delta.x^2 + x.
\sum_{i \in N} \left( 
\lambda_{i}^{(k)}.(c^s_l.a_{il}^{+} +
c^r. a_{il}^{-} )+
 u_{hi}^{(k)} a_{il}
\right)
\right)
 $
\end{enumerate}
where the first four elements are dual variable and the last four ones are 
primal ones  