X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/desynchronisation-controle.git/blobdiff_plain/bc18d50297e1b6798015a28bc91eee4f7f3752ca..b41ef425e5e9f8ab1e567e38b17f250bf4a75c75:/IWCMC14/HLG.tex

diff --git a/IWCMC14/HLG.tex b/IWCMC14/HLG.tex
index 9833667..1008903 100644
--- a/IWCMC14/HLG.tex
+++ b/IWCMC14/HLG.tex
@@ -1,56 +1,45 @@
-Let us first the basic recalls of the~\cite{HLG09} article.
-
-
-The  precise the context of video sensor network as represeted for instance 
-in figure~\ref{fig:sn}.
-
 \begin{figure}
 \begin{center}
-\includegraphics[scale=0.5]{reseau.png}
+\includegraphics[scale=0.3]{reseau.png}
+
+\begin{scriptsize}
+An example of a sensor network ofsize 10. All nodes are video sensor 
+except the 5 and the 9 one which is the sink.
+\JFC{reprendre la figure, trouver un autre titre}
+\end{scriptsize} 
+
 \caption{SN with 10 sensor}\label{fig:sn}.
 \end{center}
 \end{figure} 
 
-
-Let us give a formalisation of such a wideo network sensor.
-We start with the flow formalising:
-
-The video sensor network is represented as a strongly 
-connected oriented labelled graph. 
+Let us first recall  the basics of the~\cite{HLG09} article.
+The video sensor network is memorized as a connected non oriented 
+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
-$L$ .  
-This boolean information is stored as a  
+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
+$L$.
+Figure~\ref{fig:sn} gives an example of such a network.
+  
+This link 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.$.
+$a_{il}$ is  1 if $l$ starts with $i$, is -1 if  $l$ ends width $i$ 
+and  0 otherwise.
 
 
 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 
-  $$
-\eta_{hi} = 
-\left\{
-    \begin{array}{rl}
-      R_h & \textrm{si $i$ est $h$} \\
-      -R_h & \textrm{si $i$ est le puits} \\
-      0 & \textrm{sinon}
-    \end{array}
-  \right.$$
+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  node $i$, 
+for the  session initiated by $h$. More precisely, we have 
+$ \eta_{hi}$ is equal to $ R_h$ if $i$ is $h$,
+is equal to $-R_h$ if $i$ is the sink, and $0$ otherwise.
   
 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} $.
@@ -62,7 +51,7 @@ 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 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. 
@@ -81,24 +70,135 @@ The objective is thus to find $R$, $x$, $P_s$  which minimize
 \item $P_{sh} > 0,\forall h \in V$
 \end{enumerate}
 
-
-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 
+To achieve this optimizing goal 
+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:
+$$
+\begin{array}{l}
+P_{si}+ \sum_{l \in L}a_{il}^{+}.c^s_l.\left( \sum_{h \in V}x_{hl} \right) \\
+\qquad + 
+ \sum_{l \in L} a_{il}^{-}.c^r.\left( \sum_{h \in V}x_{hl} \right) \leq q.B_i, \forall i \in N
+\end{array}
+$$
+
+
+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 
+\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 $l \in L$.
+
+\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) $
+\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 
+  $\begin{array}{l}
+   \lambda_{i}^{(k+1)} = \lambda_{i}^{(k)} - \theta^{(k)}.\left( 
+    q^{(k)}.B_i \right.\\
+  \qquad\qquad\qquad -\sum_{l \in L}a_{il}^{+}.c^s_l.\left( \sum_{h \in V}x_{hl}^{(k)} \right)   \\
+  \qquad\qquad\qquad  - \left. \sum_{l \in L} a_{il}^{-}.c^r.\left( \sum_{h \in V}x_{hl}^{(k)} \right) - P_{si}^{(k)}  \right)
+\end{array}
+$
 
-$$P_{si}+ \sum_{l \in L}a_{il}^{+}.c^s_l.\left( \sum_{h \in V}x_{hl} \right) + 
-\sum_{l \in L} a_{il}^{-}.c^r.\left( \sum_{h \in V}x_{hl} \right) \leq q.B_i, \forall i \in N$$
-
-\JFC{VÃ©rifier l'inÃ©quation prÃ©cÃ©dente}  
+\item 
+$w_l^{(k+1)} = w_l^{(k+1)} +  \theta^{(k)}. \left( \sum_{i \in N} a_{il}.q_i^{(k)} \right)$
 
-The authors then aplly a dual based approach with Lagrange multiplier 
-to solve such a problem.
 
+\item 
+$\theta^{(k)} = \omega / t^{1/2}$
 
+ \item 
+$q_i^{(k)} = \arg\min_{q_i>0}
+\left(
+q_i^2 + q_i. 
+\left(
+\sum_{l \in L } a_{il}w_l^{(k)}-
+\lambda_i^{(k)}B_i
+\right)
+\right)$
 
+\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)
+$
 
-\inputFrameb{Formulation simplifiÃ©e}{formalisationsimplifiee}
\ No newline at end of file
+\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)} =
+\begin{array}{l}
+\arg \min_{x \geq 0}
+\left(
+\delta.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}
+$
+\end{enumerate}
+where the first four elements are dual variable and the last four ones are 
+primal ones.
\ No newline at end of file
