correction HLG

diff --git a/IWCMC14/HLG.tex b/IWCMC14/HLG.tex
index ff12b04fa9d612566bf61cd45a0eb129bcb9809d..7daf581a2d6899a0f9af1a2ce0b386b45153c9de 100644 (file)
--- a/IWCMC14/HLG.tex
+++ b/IWCMC14/HLG.tex
@@ -1,24 +1,30 @@
-\begin{figure}
+\begin{figure*}

 \begin{center}
 \begin{center}

-\includegraphics[scale=0.3]{reseau.png}

 
 

+\includegraphics[scale=0.2]{SensorNetwork.png}

 \begin{scriptsize}
 \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}.
+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 lircles 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 SN of size 10}\label{fig:sn}.

 \end{center}
 \end{center}

-\end{figure} 
+\end{figure*} 

 
 Let us first recall  the basics of the~\cite{HLG09} article.
 The video sensor network is memorized as a connected non oriented 
 
 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. 
+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 
 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
+send a message to $j$, \textit{i. e.}, the distance betwween 
+$i$ 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.
   
 $L$.
 Figure~\ref{fig:sn} gives an example of such a network.
   


@@ -27,30 +33,44 @@ matrix $A=(a_{il})_{i \in N, l \in L}$,
 where 
 $a_{il}$ is  1 if $l$ starts with $i$, is -1 if  $l$ ends width $i$ 
 and  0 otherwise.
 where 
 $a_{il}$ 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 thus $R_h$, $R_h \geq 0$,
 be the encoding rate of  video sensor $h$, $h \in V$.  
 
 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  node $i$, 
-for the  session initiated by $h$. More precisely, we have 
+Let $\eta_{hi}$ be the  rate inside the  node $i$ 
+of the production that has beeninitiated 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.
   
 $ \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 emmission 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 foolows:  $c_l^s = \alpha +\beta.d_l^{n_p} $ wehre 
+$d_l$ 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 + 


@@ -94,7 +114,7 @@ by the objective of reducing
 \label{eq:obj2}
 \end{equation}
 where $\delta$ is a regularisation factor.
 \label{eq:obj2}
 \end{equation}
 where $\delta$ is a regularisation factor.

-This indeed introduces quadratic fonctions on variables $x_{hl}$ and 
+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 
 $R_{h}$ and makes some of the functions strictly convex.
 
 The authors then apply a classical dual based approach with Lagrange multiplier 


@@ -130,7 +150,7 @@ c^r. a_{il}^{-} ) \right.\\
 \end{equation}
 
 The proposed algorithm iteratively computes the following variables 
 \end{equation}
 
 The proposed algorithm iteratively computes the following variables 

-untill the variation of the dual function is less than a given threshold.
+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) $
 \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) $


@@ -139,7 +159,7 @@ $v_{h}^{(k+1)}= \max\left\{0,v_{h}^{(k)} -  \theta^{(k)}.\left( R_h^{(k)} - \dfr
 \item 
   $\begin{array}{l}
    \lambda_{i}^{(k+1)} = \max\left\{0, \lambda_{i}^{(k)} - \theta^{(k)}.\left( 
 \item 
   $\begin{array}{l}
    \lambda_{i}^{(k+1)} = \max\left\{0, \lambda_{i}^{(k)} - \theta^{(k)}.\left( 

-    q^{(k)}.B_i \right. \left.\\
+    q^{(k)}.B_i \right. \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.\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}
   \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.\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}


@@ -168,7 +188,7 @@ P_{sh}^{(k)}
 =
 \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)
 $
 
 \right)
 $
 

diff --git a/IWCMC14/argmin.tex b/IWCMC14/argmin.tex
index 0f647807bb54d847e6df444cf4af4f5cce1bb866..47eae44f0343734a9f8e2a5366db9f0ca9283bbd 100644 (file)
--- a/IWCMC14/argmin.tex
+++ b/IWCMC14/argmin.tex
@@ -1,5 +1,5 @@
 The approach detailed previously requires to compute 
 The approach detailed previously requires to compute 

-the variable that minimizes four convex fonctions
+the variable that minimizes four convex functions

 on constraint domains, one  
 for each primal variable $q_i$, $P_{sh}$, $R_h$, and $x_{hl}$,
 Among state of the art for this optimization step, there is
 on constraint domains, one  
 for each primal variable $q_i$, $P_{sh}$, $R_h$, and $x_{hl}$,
 Among state of the art for this optimization step, there is

diff --git a/IWCMC14/convergence.tex b/IWCMC14/convergence.tex
index ea4d4c223a2e597bf7f319f08c9dd60b8cfb30b2..13b3b2d6a14c0a7394fd2df507baba0a2a0c472e 100644 (file)
--- a/IWCMC14/convergence.tex
+++ b/IWCMC14/convergence.tex
@@ -1,8 +1,8 @@
-Let us first have a discussion on the stop criterion of the citted algorithm.
+Let us first have a discussion on the stop criterion of the cited algorithm.

 We claim that even if the variation of the dual function is less than a given 
 We claim that even if the variation of the dual function is less than a given 

-threeshold, this does not ensure that the lifetime has been maximized.
+threshold, this does not ensure that the lifetime has been maximized.

 Minimizing a function on a multiple domain (as the dual function)
 Minimizing a function on a multiple domain (as the dual function)

-may indeed easilly fall into a local trap because some of introduced 
+may indeed easily fall into a local trap because some of introduced 

 variables may lead to uniformity of the output.
 
 \begin{figure}
 variables may lead to uniformity of the output.
 
 \begin{figure}


@@ -11,7 +11,7 @@ variables may lead to uniformity of the output.
   \label{fig:convergence:scatterplot}
 \end{figure}  
 
   \label{fig:convergence:scatterplot}
 \end{figure}  
 

-Experimentations have indeed shown that even if the dual
+Experiments have indeed shown that even if the dual

 function seems to be constant 
 (variations between two evaluations of this one is less than $10^{-5}$) 
 not all the $q_i$ have the same value.
 function seems to be constant 
 (variations between two evaluations of this one is less than $10^{-5}$) 
 not all the $q_i$ have the same value.


@@ -20,15 +20,15 @@ a scatter plot.
 
 The maximum amplitude rate  of the sequence of $q_i$ --which is 
 $\frac{\max_{i \in N} q_i} {\min_{i \in N}q_i}-1$--
 
 The maximum amplitude rate  of the sequence of $q_i$ --which is 
 $\frac{\max_{i \in N} q_i} {\min_{i \in N}q_i}-1$--

-is represented in $y$-coordonates 
+is represented in $y$-coordinate 

  with respect to the
  with respect to the

-value of the threeshold for dual function that is represented in 
-$x$-coordonates.
+value of the threshold for dual function that is represented in 
+$x$-coordinate.

 This figure shows that a very small threshold is a necessary condition, but not 
 a sufficient criteria to observe convergence of $q_i$.
 
 In the following, we consider the system are $\epsilon$-stable  if both 
 This figure shows that a very small threshold is a necessary condition, but not 
 a sufficient criteria to observe convergence of $q_i$.
 
 In the following, we consider the system are $\epsilon$-stable  if both 

-maximum amplitude rate and the dual function are less than a threeshold 
+maximum amplitude rate and the dual function are less than a threshold 

 $\epsilon$.
 
 
 $\epsilon$.
 
 

diff --git a/IWCMC14/convexity.tex b/IWCMC14/convexity.tex
index 9ca786acc44206dd5d9bff60a4e6529f771d5a2f..a663986869692dc80733037033e9241646508fe9 100644 (file)
--- a/IWCMC14/convexity.tex
+++ b/IWCMC14/convexity.tex
@@ -8,25 +8,40 @@ P_{sh}^{(k)}
 v_h^{(k)}.\dfrac{\ln(\sigma^2/D_h)}{\gamma p ^{2/3}} + \lambda_h^{(k)}p
 \right)
 $.
 v_h^{(k)}.\dfrac{\ln(\sigma^2/D_h)}{\gamma p ^{2/3}} + \lambda_h^{(k)}p
 \right)
 $.

-The function inside the $\arg \min$ is stricly convex if and only if 
-$\lamda_h$ is not null. This asymptotic configuration may arrise due to 
+The function inside the $\arg \min$ is strictly convex if and only if 
+$\lambda_h$ is not null. This asymptotic configuration may arise due to 

 the definition of $\lambda_i$. Worth, in this case,  the function is 
 the definition of $\lambda_i$. Worth, in this case,  the function is 

-stricly decreasing and the minimal value is obtained when $p$ is the infinity.
+strictly decreasing and the minimal value is obtained when $p$ is the infinity.

 
 To prevent this configuration, we replace the objective function given 
 in equation~(\ref{eq:obj2}) by 
 \begin{equation}
 \sum_{i \in N }q_i^2 + 
 
 To prevent this configuration, we replace the objective function given 
 in equation~(\ref{eq:obj2}) by 
 \begin{equation}
 \sum_{i \in N }q_i^2 + 

-\delta_x \sum_{h \in V, l \in L } .x_{hl}^2 
-+ \delta_r\sum_{h \in V }\delta.R_{h}^2
-+ \delta_p\sum_{h \in V }\delta.P_{sh}^{\frac{8}{3}}.
+\delta_x \sum_{h \in V, l \in L } x_{hl}^2 
++ \delta_r\sum_{h \in V }R_{h}^2
++ \delta_p\sum_{h \in V }P_{sh}^{\frac{8}{3}}.

 \label{eq:obj2}
 \end{equation}
 In this equation we have first introduced new regularisation factors
 (namely $\delta_x$, $\delta_r$, and $\delta_p$)
 instead of the sole $\delta$.  
 This allows to  further study the influence of each modification separately.
 \label{eq:obj2}
 \end{equation}
 In this equation we have first introduced new regularisation factors
 (namely $\delta_x$, $\delta_r$, and $\delta_p$)
 instead of the sole $\delta$.  
 This allows to  further study the influence of each modification separately.

-Next, the introduction of the rationnal exponent is motivated by the goal of 
-providing a stricly convex function.
+Next, the introduction of the rational exponent is motivated by the goal of 
+providing a strictly convex function.
+
+Let us now verify that the induced function is convex.
+Let $f: \R^{+*} \rightarrow \R$ such that $
+f(p)= v_h.\dfrac{\ln(\sigma^2/D_h)}{\gamma p^{2/3}} + \lambda_h p
++ \delta_p p^{8/3}$. This function is differentiable and 
+for any $x \in \R^{+*}$ and we have
+$$
+\begin{array}{rcl}
+f'(p) &=& -2/3.v_h.\dfrac{\ln(\sigma^2/D_h)}{\gamma p^{5/3}} + \lambda_h + 
+8/3.\delta_p p^{5/3} \\
+&& \dfrac {8/3.\delta_p p^{10/3} + \lambda_h p^{5/3} -2/3.v_h\ln(\sigma^2/D_h)  }{p^{5/3}}
+\end{array}
+$$
+which is positive if and only if the numerator is.
+Provided $p^{5/3}$ is replaced by $P$, we have a quadratic function which is strictly convex, for any value of $\lambda_h$. 

 
   
\ No newline at end of file
 
   
\ No newline at end of file

diff --git a/IWCMC14/main.tex b/IWCMC14/main.tex
index b887bc2170687773ae6d194fed29f068254e1767..67018d3a48784ab792850e9e1e6f56027559e571 100644 (file)
--- a/IWCMC14/main.tex
+++ b/IWCMC14/main.tex
@@ -90,15 +90,15 @@
 \input{convexity}
 \JFC{On doit parler du probleme pour Ps}
 
 \input{convexity}
 \JFC{On doit parler du probleme pour Ps}
 

-\subsection{Directly retreiving the argmin}
+\subsection{Directly retrieving the argmin}

 \input{argmin}
 
 
 
 
 \input{argmin}
 
 
 
 

-\subsection{Fault-Tolerence Considerations}
+\subsection{Fault-Tolerance Considerations}
+

 
 

-\

 
 
 \section{Discussion and Future Work}
 
 
 \section{Discussion and Future Work}