Let us first the basic recalls of the~\cite{HLG09} article.
-The precise the context of video sensor network as represeted for instance
+The precise the context of video sensor network as represented for instance
in figure~\ref{fig:sn}.
\begin{figure}
\end{figure}
-Let us give a formalisation of such a wideo network sensor.
+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.
-Furtheremore, there is an edge from $i$ to $j$ if $i$ can
-send a mesage to $j$. The set of all edges is further denoted as
+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 $A=(a_{il})_{i \in N, l \in L}$,
\eta_{hi} =
\left\{
\begin{array}{rl}
- R_h & \textrm{si $i$ est $h$} \\
- -R_h & \textrm{si $i$ est le puits} \\
- 0 & \textrm{sinon}
+ R_h & \textrm{if $i$ is $h$} \\
+ -R_h & \textrm{if $i$ is the sink} \\
+ 0 & \textrm{otherwise}
\end{array}
\right.$$
We are then left to focus on the flows in this network.
Let $x_{hl}$, $x_{hl}\geq 0$, be the flow inside the edge $l$ that
-issued from the $h$ sesssion and
+issued from the $h$ session and
let $y_l = \sum_{h \in V}x_{hl} $ the sum of all the flows inside $l$.
Thus, what is produced inside the $i^{th}$ sensor for session $h$
is $ \eta_{hi} = \sum_{l \in L }a_{il}x_{hl} $.
The initial energy of the $i$ node is $B_i$.
-The overall consumed powed of the $i$ node is
+The overall consumed power of the $i$ node is
$P_{si}+ P_{ti} + P_{ri}=
P_{si}+ \sum_{l \in L}a_{il}^{+}.c^s_l.y_l +
\sum_{l \in L} a_{il}^{-}.c^r.y_l \leq q.B_i.
To achieve a local optimisation, the problem is translated into an
-equivalent one:
-
-\begin{itemize}
-The objective is thus to find $R$, $x$, $P_s$ which minimize
+equivalent one: find $R$, $x$, $P_s$ which minimize
$\sum_{i \in N }q_i^2$ with the same set of constraints, but
item \ref{itm:q}, which is replaced by:
$$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$$
-\JFC{Vérifier l'inéquation précédente}
-The authors then aplly a dual based approach with Lagrange multiplier
+The authors then apply a dual based approach with Lagrange multiplier
to solve such a problem.
+They first introduce dual variables
+$u_{hi}$, $v_{h}$, $\lambda_{i}$, and $w_l$ for any
+$h \in V$,$ i \in N$, and $l \in L$.
+They next replace the objective of reducing $\sum_{i \in N }q_i^2$
+by the objective of reducing
+$$
+\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
+$w_l^{(k+1)} = w_l^{(k+1)} + \theta^{(k)}. \left( \sum_{i \in N} a_{il}.q_i^{(k)} \right)$
-\inputFrameb{Formulation simplifiée}{formalisationsimplifiee}
\ No newline at end of file
+\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
\ No newline at end of file
