beaucoup decourbes

[desynchronisation-controle.git] / IWCMC14 / HLG.tex

diff --git a/IWCMC14/HLG.tex b/IWCMC14/HLG.tex
index 456b3693db23dd590767c1a42d423efb813debaa..86ebbc2a9ad9bc82de3a0f89f4e34822aa6da6ed 100644 (file)
--- a/IWCMC14/HLG.tex
+++ b/IWCMC14/HLG.tex
@@ -1,157 +1,211 @@
-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{figure*}

 \begin{center}
 \begin{center}

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

 
 

+\includegraphics[scale=0.2]{SensorNetwork.png}
+\begin{scriptsize}
+
+An example of a sensor network of size 10. 
+All nodes are video sensors (depicted as small discs)
+except the 9 one which is the sink (depicted as a rectangle). 
+Large circles represent the maximum 
+transmission range which is set to 20 in a square region which is 
+$50 m \times  50 m$.
+\end{scriptsize} 
+\caption{Illustration of a Sensor Network of size 10}\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. 
+Let us first recall  the basics of the~\cite{HLG09} article.
+The video sensor network is memorized as a connected non oriented 
+graph. 

 In this one, 
 In this one, 

-the nodes, in a set $N$ are sensors, links, or the sink.
+the nodes, in a set $N$, are sensors, links, or the sink.

 Furthermore, there is an edge from $i$ to $j$ if $i$ can 
 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  
+send a message to $j$, \textit{i. e.}, the distance between 
+$i$ and $j$ is less than a given maximum 
+transmission range.
+All the possible edges are stored into a sequence  
+$L$.
+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 
 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.
+Moreover, the outgoing  links(resp. the incoming links) are represented 
+with the $A^+$ matrix (res. with the $A^-$ matrix), whose elements are defined:
+$a_{il}^+$ (resp. $a_{il}^-$) is 1  if the link $l$ is an outgoing link from $i$ 
+(resp an incoming link into $i$) and 0 otherwise. 

 
 Let $V \subset N $ be the set of the video sensors of $N$.
 
 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 
-  $$
-\eta_{hi} = 
-\left\{
-    \begin{array}{rl}
-      R_h & \textrm{if $i$ is $h$} \\
-      -R_h & \textrm{if $i$ is the sink} \\
-      0 & \textrm{otherwise}
-    \end{array}
-  \right.$$
+Let thus $R_h$, $R_h \geq 0$,
+be the encoding rate of  video sensor $h$, $h \in V$.  
+Let $\eta_{hi}$ be the  rate inside the  node $i$ 
+of the production that has been 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 us focus on the flows in this network.

 Let $x_{hl}$, $x_{hl}\geq 0$, be the flow inside the edge $l$ that 
 Let $x_{hl}$, $x_{hl}\geq 0$, be the flow inside the edge $l$ that 

-issued from the $h$ session and 
+issued from the node $h$  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 $P_{si}$, $P_{si} > 0$.
 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 emission distortion  of the $i$ node is 
+$\sigma^2 e^{-\gamma . R_i.P_{si}^{}2/3}$
+where $\sigma^2$ is the average input variance and
+$\gamma$ is the encoding efficiency coefficient.
+This distortion  
+is bounded by a constant value $D_h$.

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

-
+The transmission consumed power of node $i$ is  
+$P_{ti} = c_l^s.y_l$ where  $c_l^s$ is the transmission energy
+consumption cost of link $l$, $l\in L$. This cost is defined 
+as follows:  $c_l^s = \alpha +\beta.d_l^{n_p} $ where 
+$d_l$ represents the distance of the link $l$,
+$\alpha$, $\beta$, and $n_p$ are constant. 
+The reception consumed power of node $i$ is  
+$P_{ri} = c^r \sum_{l \in L } a_{il}^-.y_l$ 
+where  $c^r$ is a reception energy consumption cost.

 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 + 
 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
+\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 maximizes
+the network lifetime $T_{\textit{net}}$, or equivalently which minimizes
+$q=1/{T_{\textit{net}}}$. 
+Let $B_i$ is the initial energy in node $i$.
+One have the equivalent objective to find $R$, $x$, $P_s$ which minimizes
+$q^2$
+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 + 
 \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$
+c^r.\sum_{l \in L} a_{il}^{-}.y_l \leq q.B_i, \forall i \in N$
+\item $\sum_{i \in N} a_{il}q_i = 0, \forall l \in L$ 

 \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}
 
 \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 
+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:
 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:

+$$
+\begin{array}{l}
+P_{si}+ \sum_{l \in L}a_{il}^{+}.c^s_l.\left( \sum_{h \in V}x_{hl} \right) \\
+\qquad + 
+ \sum_{l \in L} a_{il}^{-}.c^r.\left( \sum_{h \in V}x_{hl} \right) \leq q_i.B_i, \forall i \in N
+\end{array}
+$$
+and where the following constraint is added
+$q_i > 0, \forall i \in N$.

 
 

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

 
 
 
 

-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 $\sum_{i \in N }q_i^2$
+They thus replace the objective of reducing
+$\sum_{i \in N }q_i^2$

 by the objective of reducing 
 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 
 \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 functions 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} \left( q_i^2 + q_i.  \left(
+\sum_{l \in L } a_{il}w_l-
+\lambda_iB_i
+\right)\right) \\
++ \sum_{h \in V} \left(
+v_h.\dfrac{\ln(\sigma^2/D_h)}{\gamma P_{sh} ^{2/3}} + \lambda_h P_{sh} \right)\\
++ \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} \left(
+\delta R_{h}^2 
+-v_h.R_{h} - \sum_{i \in N} u_{hi}\eta_{hi}\right)
+\end{array}
+\label{eq:dualFunction}
+\end{equation}

 
 The proposed algorithm iteratively computes the following variables 
 
 The proposed algorithm iteratively computes the following variables 

-
+until 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{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)
+  $\begin{array}{l}
+   \lambda_{i}^{(k+1)} = \max\left\{0, \lambda_{i}^{(k)} - \theta^{(k)}.\left( 
+    q^{(k)}.B_i - P_{si}^{(k)} \right. \right.\\
+  \qquad\qquad\qquad -\sum_{l \in L}a_{il}^{+}.c^s_l. \sum_{h \in V}x_{hl}^{(k)}    \\
+  \qquad\qquad\qquad  - \left.\left. c^r.\sum_{l \in L} a_{il}^{-}. \sum_{h \in V}x_{hl}^{(k)} \right)   \right\}

 \end{array}
 $
 
 \item 
 \end{array}
 $
 
 \item 

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

 
 
 \item 
 
 
 \item 

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

 
  \item 
 
  \item 

-$q_i^{(k)} = \arg\min_{q_i>0}
+$q_i^{(k+1)} = \arg\min_{q_i>0}

 \left(
 \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)$
 
 \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)} 
+P_{sh}^{(k+1)} 

 =
 \arg \min_{p > 0} 
 \left(
 =
 \arg \min_{p > 0} 
 \left(

-v_h^{(k)}.\dfrac{\ln(\sigma^2/D_h)}{\gamma p ^{2/3}} + \lambda_h^{(k)}p
+v_h^{(k)}.\dfrac{\ln(\sigma^2/D_h)}{\gamma p^{2/3}} + \lambda_h^{(k)}p

 \right)
 $
 
 \item 
 $
 \right)
 $
 
 \item 
 $

-R_h^{(k)}
+R_h^{(k+1)}

 =
 \arg \min_{r \geq 0 }
 \left(
 =
 \arg \min_{r \geq 0 }
 \left(


@@ -161,17 +215,20 @@ R_h^{(k)}
 $
 \item 
 $
 $
 \item 
 $

-x_{hl}^{(k)} =
+x_{hl}^{(k+1)} =
+\begin{array}{l}

 \arg \min_{x \geq 0}
 \left(
 \arg \min_{x \geq 0}
 \left(

-\delta.x^2 + x.
+\delta.x^2  \right.\\
+\qquad \qquad + x.

 \sum_{i \in N} \left( 
 \lambda_{i}^{(k)}.(c^s_l.a_{il}^{+} +
 \sum_{i \in N} \left( 
 \lambda_{i}^{(k)}.(c^s_l.a_{il}^{+} +

-c^r. a_{il}^{-} )+
- u_{hi}^{(k)} a_{il}
+c^r. a_{il}^{-} ) \right.\\
+\qquad \qquad\qquad \qquad +
+\left.\left. u_{hi}^{(k)} a_{il}

 \right)
 \right)
 \right)
 \right)

- $
+\end{array}
+$

 \end{enumerate}
 \end{enumerate}

-where the first four elements are dual variable and the last four ones are 
-primal ones  
\ No newline at end of file
+for any $h \in V$, $i \in N$, and $l \in L$.