quelques corrections en +

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

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 $\lambda_h$. 

 
   
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