X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/desynchronisation-controle.git/blobdiff_plain/1216fb04e58a581d9a4d2e77355aaf47bf989447..HEAD:/IWCMC14/convexity.tex

diff --git a/IWCMC14/convexity.tex b/IWCMC14/convexity.tex
index 9ca786a..d88481c 100644
--- a/IWCMC14/convexity.tex
+++ b/IWCMC14/convexity.tex
@@ -8,25 +8,68 @@ P_{sh}^{(k)}
 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 definition of $\lambda_i$. Worth, in this case,  the function is 
-stricly decreasing and the minimal value is obtained when $p$ is the infinity.
+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 
 \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}}.
-\label{eq:obj2}
+\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: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 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.
+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.
+
+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
+$$
+\begin{array}{rcl}
+f'(p) &=& -2/3.v_h.\dfrac{\ln(\sigma^2/D_h)}{\gamma p^{5/3}} + \lambda_h + 
+8/3.\delta_p p^{5/3} \\
+& = & \dfrac {8/3.\delta_p p^{10/3} + \lambda_h p^{5/3} -2/3\gamma.v_h\ln(\sigma^2/D_h)  }{p^{5/3}}
+\end{array}
+$$
+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 $\lambda_h$ since the discriminant 
+is positive. 
+
+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 threshold with 
+the two approaches: either the original one ore the
+one which is convex guarantee.
+
+The Figure~\ref{Fig:convex} summarizes the average number of convergence 
+iterations for each treshold value. As we can see, even if this new 
+enhanced method introduces new calculus, 
+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 Guarantee Approaches}\label{Fig:convex}
+\end{figure*} 
+
 
-  
\ No newline at end of file