beaucoup decourbes

diff --git a/IWCMC14/HLG.tex b/IWCMC14/HLG.tex
index 998f94b5aca6f97d90777f699b241e382fd812f8..86ebbc2a9ad9bc82de3a0f89f4e34822aa6da6ed 100644 (file)
--- a/IWCMC14/HLG.tex
+++ b/IWCMC14/HLG.tex
@@ -7,7 +7,7 @@
 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). 
 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 
+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} 
 transmission range which is set to 20 in a square region which is 
 $50 m \times  50 m$.
 \end{scriptsize} 


@@ -21,7 +21,7 @@ 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 $j$, \textit{i. e.}, the distance betwween 
+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  
 $i$ and $j$ is less than a given maximum 
 transmission range.
 All the possible edges are stored into a sequence  


@@ -42,7 +42,7 @@ 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  rate inside the  node $i$ 
 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 beeninitiated by $h$. More precisely, we have 
+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.
   
 $ \eta_{hi}$ is equal to $ R_h$ if $i$ is $h$,
 is equal to $-R_h$ if $i$ is the sink, and $0$ otherwise.
   


@@ -55,7 +55,7 @@ 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 encoding power of the $i$ node is $P_{si}$, $P_{si} > 0$.

-The emmission distortion  of the $i$ node is 
+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.
 $\sigma^2 e^{-\gamma . R_i.P_{si}^{}2/3}$
 where $\sigma^2$ is the average input variance and
 $\gamma$ is the encoding efficiency coefficient.


@@ -65,7 +65,7 @@ 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 
 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} $ where 
+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  
 $d_l$ represents the distance of the link $l$,
 $\alpha$, $\beta$, and $n_p$ are constant. 
 The reception consumed power of node $i$ is  


@@ -110,7 +110,7 @@ P_{si}+ \sum_{l \in L}a_{il}^{+}.c^s_l.\left( \sum_{h \in V}x_{hl} \right) \\
 \end{array}
 $$
 and where the following constraint is added
 \end{array}
 $$
 and where the following constraint is added

-$$ $q_i > 0, \forall i \in N  $$
+$q_i > 0, \forall i \in N$.

 
 
 
 
 
 


@@ -131,17 +131,17 @@ 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 
 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 $l \in L$.
+$h \in V$, $ i \in N$, and $l \in L$.

 
 \begin{equation}
 \begin{array}{l}
 L(R,x,P_{s},q,u,v,\lambda,w)=\\ 
 
 \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)}-
+\sum_{i \in N} \left( q_i^2 + q_i.  \left(
+\sum_{l \in L } a_{il}w_l-

 \lambda_iB_i
 \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}\\
+\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.\\
 + \sum_{h \in V} \sum_{l\in L}
 \left(
 \delta.x_{hl}^2  \right.\\


@@ -153,10 +153,11 @@ c^r. a_{il}^{-} ) \right.\\
 \left.\left. u_{hi} a_{il}
 \right)
 \right)\\
 \left.\left. u_{hi} a_{il}
 \right)
 \right)\\

- + \sum_{h \in V}
+ + \sum_{h \in V} \left(

 \delta R_{h}^2 
 \delta R_{h}^2 

--v_h.R_{h} - \sum_{i \in N} u_{hi}\eta_{hi}
+-v_h.R_{h} - \sum_{i \in N} u_{hi}\eta_{hi}\right)

 \end{array}
 \end{array}

+\label{eq:dualFunction}

 \end{equation}
 
 The proposed algorithm iteratively computes the following variables 
 \end{equation}
 
 The proposed algorithm iteratively computes the following variables 


@@ -169,21 +170,21 @@ $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. \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\}
+    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(
 q_i^2 + q_i. 
 \left(
 \left(
 q_i^2 + q_i. 
 \left(


@@ -194,7 +195,7 @@ q_i^2 + q_i.
 
 \item \label{item:psh} 
 $
 
 \item \label{item:psh} 
 $

-P_{sh}^{(k)} 
+P_{sh}^{(k+1)} 

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


@@ -204,7 +205,7 @@ $
 
 \item 
 $
 
 \item 
 $

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

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


@@ -214,7 +215,7 @@ R_h^{(k)}
 $
 \item 
 $
 $
 \item 
 $

-x_{hl}^{(k)} =
+x_{hl}^{(k+1)} =

 \begin{array}{l}
 \arg \min_{x \geq 0}
 \left(
 \begin{array}{l}
 \arg \min_{x \geq 0}
 \left(


@@ -230,5 +231,4 @@ c^r. a_{il}^{-} ) \right.\\
 \end{array}
 $
 \end{enumerate}
 \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
+for any $h \in V$, $i \in N$, and $l \in L$. 

diff --git a/IWCMC14/argmin.tex b/IWCMC14/argmin.tex
index 47eae44f0343734a9f8e2a5366db9f0ca9283bbd..be684d45b5657674491e9625fe56d969c0c9ec6e 100644 (file)
--- a/IWCMC14/argmin.tex
+++ b/IWCMC14/argmin.tex
@@ -7,21 +7,25 @@ the  L-BFGS-B algorithm~\cite{byrd1995limited},
 the truncated Newton algorithm, 
 the Constrained Optimization BY Linear Approximation (COBYLA) method~\cite{ANU:1770520}.
 However, all these methods suffer from being iterative approaches 
 the truncated Newton algorithm, 
 the Constrained Optimization BY Linear Approximation (COBYLA) method~\cite{ANU:1770520}.
 However, all these methods suffer from being iterative approaches 

-and need many steps of computation to obtain an approximation 
-of the minimal value. This approach is dramatic whilst the objective is to 
-reduce all the computation steps. 
+each iteration including  many steps of computation to obtain an approximation 
+of the minimal value. 
+This approach is dramatic since the objective is to 
+reduce all the computation steps to increase the network lifetime. 

   
 A closer look to each function that has to be minimized shows that it is
 differentiable and the minimal value can be computed in only one step. 
 The table~\ref{table:min} presents these minimal value for each primal
 variable.
   
 A closer look to each function that has to be minimized shows that it is
 differentiable and the minimal value can be computed in only one step. 
 The table~\ref{table:min} presents these minimal value for each primal
 variable.

+Thanks to this formal calculus, computing the 
+new iterate of each primal variable  only requires 
+one computation step.

 
 \begin{table*}[t]
 $$
 \begin{array}{|l|l|l|}
 \hline
 
 \begin{table*}[t]
 $$
 \begin{array}{|l|l|l|}
 \hline

-q_i^{(k)} &
- \arg\min_{q_i>0}
+q_i^{(k+1)} &
+ \arg\min_{q>0}

 \left(
 q^2 + q. 
 \left(
 \left(
 q^2 + q. 
 \left(


@@ -30,35 +34,35 @@ q^2 + q.
 \right)
 \right)
  & 
 \right)
 \right)
  & 

-\max \left(\epsilon,\dfrac{\sum_{l \in L } a_{il}w_l^{(k)}-
-\lambda_i^{(k)}B_i}{2}\right) \\
+\max \left\{\epsilon,\dfrac{\sum_{l \in L } a_{il}w_l^{(k)}-
+\lambda_i^{(k)}B_i}{2}\right\} \\

 \hline
 \hline

-P_{sh}^{(k)}& 
+P_{sh}^{(k+1)}& 

 \arg \min_{p > 0} 
 \left(
 v_h^{(k)}.\dfrac{\ln(\sigma^2/D_h)}{\gamma p^{2/3}} + \lambda_h^{(k)}p
 + \delta_p p^{8/3}
 \right)
 &
 \arg \min_{p > 0} 
 \left(
 v_h^{(k)}.\dfrac{\ln(\sigma^2/D_h)}{\gamma p^{2/3}} + \lambda_h^{(k)}p
 + \delta_p p^{8/3}
 \right)
 &

-\max \left(\epsilon,
+\max \left\{\epsilon,

 \left(
 \dfrac{
 -\lambda_h^{(k)} + \sqrt{(\lambda_h^{(k)})^2 + \dfrac{64}{9}\alpha}
 }{\frac{16}{3}\delta_p}
 \right)^{\frac{3}{5}}
 \left(
 \dfrac{
 -\lambda_h^{(k)} + \sqrt{(\lambda_h^{(k)})^2 + \dfrac{64}{9}\alpha}
 }{\frac{16}{3}\delta_p}
 \right)^{\frac{3}{5}}

-\right) \\
+\right\} \\

 \hline
 \hline

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

 &
 \arg \min_{r \geq 0 }
 \left(
 \delta_r r^2 
 -v_h^{(k)}.r - \sum_{i \in N} u_{hi}^{(k)} \eta_{hi}
 \right) &
 &
 \arg \min_{r \geq 0 }
 \left(
 \delta_r r^2 
 -v_h^{(k)}.r - \sum_{i \in N} u_{hi}^{(k)} \eta_{hi}
 \right) &

-\max\left(0,\dfrac{v_h^{(k)}}{2\delta_r}\right)
+\max\left\{0,\dfrac{v_h^{(k)}}{2\delta_r}\right\}

 \\
 \hline
 \\
 \hline

-x_{hl}^{(k)} &
+x_{hl}^{(k+1)} &

 \begin{array}{l}
 \arg \min_{x \geq 0}
 \left(
 \begin{array}{l}
 \arg \min_{x \geq 0}
 \left(


@@ -73,15 +77,16 @@ c^r. a_{il}^{-} ) \right.\\
 \right)
 \end{array}
 &
 \right)
 \end{array}
 &

-\max\left(0,\dfrac{-\sum_{i \in N} \left( 
+\max\left\{0,\dfrac{-\sum_{i \in N} \left( 

 \lambda_{i}^{(k)}.(c^s_l.a_{il}^{+} +
 c^r. a_{il}^{-} ) + u_{hi}^{(k)} a_{il}
 \lambda_{i}^{(k)}.(c^s_l.a_{il}^{+} +
 c^r. a_{il}^{-} ) + u_{hi}^{(k)} a_{il}

-\right)}{2\delta_x}\right)
+\right)}{2\delta_x}\right\}

 \\
 \hline
 \end{array}
 $$
 \\
 \hline
 \end{array}
 $$

-\caption{Expression of each optimized primal variable}
+\caption{Primal Variables: Argmin and Direct Calculus}\label{table:min}

 \end{table*}
 
 \end{table*}
 

-  
\ No newline at end of file
+
+This improvement 
\ No newline at end of file

diff --git a/IWCMC14/convergence.tex b/IWCMC14/convergence.tex
index 13b3b2d6a14c0a7394fd2df507baba0a2a0c472e..b689f9900f32e5deec97eb399cd7eadd7a0e75d4 100644 (file)
--- a/IWCMC14/convergence.tex
+++ b/IWCMC14/convergence.tex
@@ -1,32 +1,42 @@
 Let us first have a discussion on the stop criterion of the cited 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
+(recalled in equation (\ref{eq:dualFunction}))
+is less than a given 

 threshold, this does not ensure that the lifetime has been maximized.
 Minimizing a function on a multiple domain (as the dual function)
 may indeed easily fall into a local trap because some of introduced 
 variables may lead to uniformity of the output.
 
 \begin{figure}
 threshold, this does not ensure that the lifetime has been maximized.
 Minimizing a function on a multiple domain (as the dual function)
 may indeed easily fall into a local trap because some of introduced 
 variables may lead to uniformity of the output.
 
 \begin{figure}

-  to be continued 
-  \caption{Relations between dual function threshold and $q_i$ convergence}
+  \begin{center}
+    \includegraphics[scale=0.5]{amplrate.png}
+  \end{center}
+  \caption{Relations between dual function variation and convergence of all the $q_i$}

   \label{fig:convergence:scatterplot}
 \end{figure}  
 
   \label{fig:convergence:scatterplot}
 \end{figure}  
 

-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.
-For instance, the Figure~\ref{fig:convergence:scatterplot} presents 
-a scatter plot.
-
-The maximum amplitude rate  of the sequence of $q_i$ --which is 
-$\frac{\max_{i \in N} q_i} {\min_{i \in N}q_i}-1$--
+To explain this, we introduce the maximum amplitude rate $\zeta$ 
+of the sequence of $q$ which is defined as  
+$$
+\dfrac{\max_{i \in N} \{q_i\}}
+{\min_{i \in N} \{q_i\}}-1.
+$$
+The Figure~\ref{fig:convergence:scatterplot} presents 
+a scatter plot between $\zeta$, which 

 is represented in $y$-coordinate 
 is represented in $y$-coordinate 

- with respect to the
+with respect to the

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

 
 

+
+Experiments 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, \textit{i. e.}, $\zeta$ is still large.
+For instance, even with a threshold set to $10^{-5}$ there still can be more than
+45\% of differences between two $q_i$. 
+To summarize, 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 threshold 
 $\epsilon$.
 In the following, we consider the system are $\epsilon$-stable  if both 
 maximum amplitude rate and the dual function are less than a threshold 
 $\epsilon$.

diff --git a/IWCMC14/convexity.tex b/IWCMC14/convexity.tex
index a663986869692dc80733037033e9241646508fe9..0f7bf7e7e6f076cf4a54ccf8c6c0db1d854c787f 100644 (file)
--- a/IWCMC14/convexity.tex
+++ b/IWCMC14/convexity.tex
@@ -10,7 +10,7 @@ v_h^{(k)}.\dfrac{\ln(\sigma^2/D_h)}{\gamma p ^{2/3}} + \lambda_h^{(k)}p
 $.
 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 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 $\lambda_h$. Worth, in this case,  the function is 

 strictly decreasing and the minimal value is obtained when $p$ is the infinity.
 
 To prevent this configuration, we replace the objective function given 
 strictly decreasing and the minimal value is obtained when $p$ is the infinity.
 
 To prevent this configuration, we replace the objective function given 


@@ -25,7 +25,7 @@ in equation~(\ref{eq:obj2}) by
 In this equation we have first introduced new regularisation factors
 (namely $\delta_x$, $\delta_r$, and $\delta_p$)
 instead of the sole $\delta$.  
 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.
+This allows to  further separately study the influence of each factor.

 Next, the introduction of the rational exponent is motivated by the goal of 
 providing a strictly convex function.
 
 Next, the introduction of the rational exponent is motivated by the goal of 
 providing a strictly convex function.
 


@@ -38,10 +38,12 @@ $$
 \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} \\
 \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}}
+& = & \dfrac {8/3\gamma.\delta_p p^{10/3} + \lambda_h p^{5/3} -2/3.v_h\ln(\sigma^2/D_h)  }{p^{5/3}}

 \end{array}
 $$
 which is positive if and only if the numerator is.
 \end{array}
 $$
 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 $\lambda_h$. 
+Provided $p^{5/3}$ is replaced by $P$, we have a quadratic function 
+which is strictly convex, for any value of $\lambda_h$ since the discriminant 
+is positive. 

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

diff --git a/IWCMC14/main.tex b/IWCMC14/main.tex
index 67018d3a48784ab792850e9e1e6f56027559e571..bd4e952938a5842877f00be1816952935fa8b315 100644 (file)
--- a/IWCMC14/main.tex
+++ b/IWCMC14/main.tex
@@ -21,6 +21,7 @@
 \newcommand{\CG}[1]{\begin{color}{blue}\textit{}\end{color}}
 
 
 \newcommand{\CG}[1]{\begin{color}{blue}\textit{}\end{color}}
 
 

+\newcommand{\R}[0]{\ensuremath{\mathbb{R}}}

 
 
 \author{
 
 
 \author{


@@ -88,7 +89,6 @@
 
 \subsection{A closer look to convexity}
 \input{convexity}
 
 \subsection{A closer look to convexity}
 \input{convexity}

-\JFC{On doit parler du probleme pour Ps}

 
 \subsection{Directly retrieving the argmin}
 \input{argmin}
 
 \subsection{Directly retrieving the argmin}
 \input{argmin}


@@ -107,7 +107,7 @@
 
 %\bibliographystyle{compj}
 \bibliographystyle{IEEEtran}
 
 %\bibliographystyle{compj}
 \bibliographystyle{IEEEtran}

-\bibliography{forensicsVer4}
+\bibliography{abbrev,biblioand}