From: Jean-François Couchot <couchot@couchot.iut-bm.univ-fcomte.fr>
Date: Tue, 3 Dec 2013 16:32:15 +0000 (+0100)
Subject: convexity modification
X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/desynchronisation-controle.git/commitdiff_plain/110a1a5d3f942917cabd4b60afa51b0ea2989229?ds=sidebyside;hp=30b003311b2d550b2d7fbe8179df2320df1acf2f

convexity modification
---

diff --git a/IWCMC14/HLG.tex b/IWCMC14/HLG.tex
index 1008903..ff12b04 100644
--- a/IWCMC14/HLG.tex
+++ b/IWCMC14/HLG.tex
@@ -138,10 +138,10 @@ untill the variation of the dual function is less than a given threshold.
 $v_{h}^{(k+1)}= \max\left\{0,v_{h}^{(k)} -  \theta^{(k)}.\left( R_h^{(k)} - \dfrac{\ln(\sigma^2/D_h)}{\gamma.(P_{sh}^{(k)})^{2/3}}   \right)\right\}$
 \item 
   $\begin{array}{l}
-   \lambda_{i}^{(k+1)} = \lambda_{i}^{(k)} - \theta^{(k)}.\left( 
-    q^{(k)}.B_i \right.\\
+   \lambda_{i}^{(k+1)} = \max\left\{0, \lambda_{i}^{(k)} - \theta^{(k)}.\left( 
+    q^{(k)}.B_i \right. \left.\\
   \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. \sum_{l \in L} a_{il}^{-}.c^r.\left( \sum_{h \in V}x_{hl}^{(k)} \right) - P_{si}^{(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\}
 \end{array}
 $
 
@@ -162,7 +162,7 @@ q_i^2 + q_i.
 \right)
 \right)$
 
-\item 
+\item \label{item:psh} 
 $
 P_{sh}^{(k)} 
 =
diff --git a/IWCMC14/convexity.tex b/IWCMC14/convexity.tex
index 0519ecb..9ca786a 100644
--- a/IWCMC14/convexity.tex
+++ b/IWCMC14/convexity.tex
@@ -1 +1,32 @@
- 
\ No newline at end of file
+In the algorithm presented in the previous section,  
+the encoding power consumption is iteratively updated with 
+$
+P_{sh}^{(k)} 
+=
+\arg \min_{p > 0} 
+\left(
+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.
+
+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}
+\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.
+
+  
\ No newline at end of file
diff --git a/IWCMC14/main.tex b/IWCMC14/main.tex
index 49d816d..b887bc2 100644
--- a/IWCMC14/main.tex
+++ b/IWCMC14/main.tex
@@ -1,4 +1,4 @@
-\documentclass[10pt]{IEEEtran}
+\documentclass{IEEEtran}
 \usepackage[utf8]{inputenc}
 \usepackage[T1]{fontenc}
 \usepackage[english]{babel}