totot

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

diff --git a/IWCMC14/convexity.tex b/IWCMC14/convexity.tex
index ff4656dcfcad14eca82c7a5283e3ac7f909478b4..d88481cf4a58773f0ded55ea3b816b4766826232 100644 (file)
--- a/IWCMC14/convexity.tex
+++ b/IWCMC14/convexity.tex
@@ -12,6 +12,10 @@ 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 $\lambda_h$. Worth, in this case,  the function is 
 strictly decreasing and the minimal value is obtained when $p$ is the infinity.
 $\lambda_h$ is not null. This asymptotic configuration may arise due to 
 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.

+Thus, the method  follows its iterative calculus
+with an arbitrarely large value for $P_{sh}^{(k)}$. This leads to 
+a convergence which is dramatically slow down.
+

 
 To prevent this configuration, we replace the objective function given 
 in equation~(\ref{eq:obj2}) by 
 
 To prevent this configuration, we replace the objective function given 
 in equation~(\ref{eq:obj2}) by 


@@ -22,7 +26,7 @@ in equation~(\ref{eq:obj2}) by
 + \delta_p\sum_{h \in V }P_{sh}^{\frac{8}{3}}.
 \label{eq:obj2p}
 \end{equation}
 + \delta_p\sum_{h \in V }P_{sh}^{\frac{8}{3}}.
 \label{eq:obj2p}
 \end{equation}

-In this equation we have first introduced new regularisation factors
+In this equation we have first introduced new regularization factors

 (namely $\delta_x$, $\delta_r$, and $\delta_p$)
 instead of the sole $\delta$.  
 This allows to  further separately study the influence of each factor.
 (namely $\delta_x$, $\delta_r$, and $\delta_p$)
 instead of the sole $\delta$.  
 This allows to  further separately study the influence of each factor.


@@ -46,27 +50,26 @@ 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. 
 
 which is strictly convex, for any value of $\lambda_h$ since the discriminant 
 is positive. 
 

-This proposed enhacement has been evaluated as follows:  
-10 tresholds $t$, such that $1E-5 \le t \le 1E-3$, have 
+This proposed enhancement has been evaluated as follows:  
+10 thresholds $t$, such that $1E-5 \le t \le 1E-3$, have 

 been selected and for each of them,  
 10 random configurations have been generated.
 For each one, we store the 
 number of iterations which is sufficient to make the dual 
 been selected and for each of them,  
 10 random configurations have been generated.
 For each one, we store the 
 number of iterations which is sufficient to make the dual 

-function variation smaller than this given treshold with 
+function variation smaller than this given threshold with 

 the two approaches: either the original one ore the
 the two approaches: either the original one ore the

-one which is convex garantee.
+one which is convex guarantee.

 
 The Figure~\ref{Fig:convex} summarizes the average number of convergence 
 
 The Figure~\ref{Fig:convex} summarizes the average number of convergence 

-iterations for each tresholdvalue. As we can see, even if this new 
+iterations for each treshold value. As we can see, even if this new 

 enhanced method introduces new calculus, 
 enhanced method introduces new calculus, 

-it only slows few down the algorithm and garantee the convexity, 
+it speeds up  the algorithm and guarantees the convexity, 

 and thus the convergence.
 and thus the convergence.

-

 \begin{figure*}
 \begin{center}
 \includegraphics[scale=0.5]{convex.png}
 \end{center}
 \begin{figure*}
 \begin{center}
 \includegraphics[scale=0.5]{convex.png}
 \end{center}

-\caption{Original Vs Convex Garantee Approaches}\label{Fig:convex}
+\caption{Original Vs Convex Guarantee Approaches}\label{Fig:convex}

 \end{figure*} 
 
 
 \end{figure*}