X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/desynchronisation-controle.git/blobdiff_plain/17a131a2d00611e1582103549a9c7d2bac2be03e..ae93ee0d91e2a8f31ec45f7b77f9e81294630f07:/IWCMC14/argmin.tex

diff --git a/IWCMC14/argmin.tex b/IWCMC14/argmin.tex
index 47eae44..6aae1e5 100644
--- a/IWCMC14/argmin.tex
+++ b/IWCMC14/argmin.tex
@@ -7,21 +7,25 @@ the  L-BFGS-B algorithm~\cite{byrd1995limited},
 the truncated Newton algorithm, 
 the Constrained Optimization BY Linear Approximation (COBYLA) method~\cite{ANU:1770520}.
 However, all these methods suffer from being iterative approaches 
-and need many steps of computation to obtain an approximation 
-of the minimal value. This approach is dramatic whilst the objective is to 
-reduce all the computation steps. 
+each iteration including  many steps of computation to obtain an approximation 
+of the minimal value. 
+This approach is dramatic since the objective is to 
+reduce all the computation steps to increase the network lifetime. 
   
 A closer look to each function that has to be minimized shows that it is
 differentiable and the minimal value can be computed in only one step. 
 The table~\ref{table:min} presents these minimal value for each primal
 variable.
+Thanks to this formal calculus, computing the 
+new iterate of each primal variable  only requires 
+one computation step.
 
 \begin{table*}[t]
 $$
 \begin{array}{|l|l|l|}
 \hline
-q_i^{(k)} &
- \arg\min_{q_i>0}
+q_i^{(k+1)} &
+ \arg\min_{q>0}
 \left(
 q^2 + q. 
 \left(
@@ -30,35 +34,35 @@ q^2 + q.
 \right)
 \right)
  & 
-\max \left(\epsilon,\dfrac{\sum_{l \in L } a_{il}w_l^{(k)}-
-\lambda_i^{(k)}B_i}{2}\right) \\
+\max \left\{\epsilon,\dfrac{\sum_{l \in L } a_{il}w_l^{(k)}-
+\lambda_i^{(k)}B_i}{2}\right\} \\
 \hline
-P_{sh}^{(k)}& 
+P_{sh}^{(k+1)}& 
 \arg \min_{p > 0} 
 \left(
 v_h^{(k)}.\dfrac{\ln(\sigma^2/D_h)}{\gamma p^{2/3}} + \lambda_h^{(k)}p
 + \delta_p p^{8/3}
 \right)
 &
-\max \left(\epsilon,
+\max \left\{\epsilon,
 \left(
 \dfrac{
--\lambda_h^{(k)} + \sqrt{(\lambda_h^{(k)})^2 + \dfrac{64}{9}\alpha}
-}{\frac{16}{3}\delta_p}
+-3\lambda_h^{(k)} + \sqrt{(3\lambda_h^{(k)})^2 + 64\delta_p/\gamma.\ln(\sigma^2/D_h)}
+}{16\delta_p}
 \right)^{\frac{3}{5}}
-\right) \\
+\right\} \\
 \hline
-R_h^{(k)}
+R_h^{(k+1)}
 &
 \arg \min_{r \geq 0 }
 \left(
 \delta_r r^2 
 -v_h^{(k)}.r - \sum_{i \in N} u_{hi}^{(k)} \eta_{hi}
 \right) &
-\max\left(0,\dfrac{v_h^{(k)}}{2\delta_r}\right)
+\max\left\{0,\dfrac{v_h^{(k)}}{2\delta_r}\right\}
 \\
 \hline
-x_{hl}^{(k)} &
+x_{hl}^{(k+1)} &
 \begin{array}{l}
 \arg \min_{x \geq 0}
 \left(
@@ -73,15 +77,16 @@ c^r. a_{il}^{-} ) \right.\\
 \right)
 \end{array}
 &
-\max\left(0,\dfrac{-\sum_{i \in N} \left( 
+\max\left\{0,\dfrac{-\sum_{i \in N} \left( 
 \lambda_{i}^{(k)}.(c^s_l.a_{il}^{+} +
 c^r. a_{il}^{-} ) + u_{hi}^{(k)} a_{il}
-\right)}{2\delta_x}\right)
+\right)}{2\delta_x}\right\}
 \\
 \hline
 \end{array}
 $$
-\caption{Expression of each optimized primal variable}
+\caption{Primal Variables: Argmin and Direct Calculus}\label{table:min}
 \end{table*}
 
-  
\ No newline at end of file
+
+This improvement 
\ No newline at end of file