X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/desynchronisation-controle.git/blobdiff_plain/fe438aa764b5d1ad7ea61512cffbb3761eff99b7..HEAD:/IWCMC14/convexity.tex?ds=inline

diff --git a/IWCMC14/convexity.tex b/IWCMC14/convexity.tex
index ff4656d..d88481c 100644
--- 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.
+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 
@@ -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}
-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.
@@ -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. 
 
-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 
-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
-one which is convex garantee.
+one which is convex guarantee.
 
 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, 
-it only slows few down the algorithm and garantee the convexity, 
+it speeds up  the algorithm and guarantees the convexity, 
 and thus the convergence.
-
 \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*} 
 
 
