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

diff --git a/IWCMC14/HLG.tex b/IWCMC14/HLG.tex
index 183e00b45dae8115defdf97e0577eabd476edc4f..ff12b04fa9d612566bf61cd45a0eb129bcb9809d 100644 (file)
--- a/IWCMC14/HLG.tex
+++ b/IWCMC14/HLG.tex
@@ -70,7 +70,6 @@ The objective is thus to find $R$, $x$, $P_s$  which minimize
 \item $P_{sh} > 0,\forall h \in V$
 \end{enumerate}
 
 \item $P_{sh} > 0,\forall h \in V$
 \end{enumerate}
 

-

 To achieve this optimizing goal 
 a local optimisation, the problem is translated into an 
 equivalent one: find $R$, $x$, $P_s$  which minimize 
 To achieve this optimizing goal 
 a local optimisation, the problem is translated into an 
 equivalent one: find $R$, $x$, $P_s$  which minimize 


@@ -84,23 +83,54 @@ P_{si}+ \sum_{l \in L}a_{il}^{+}.c^s_l.\left( \sum_{h \in V}x_{hl} \right) \\
 \end{array}
 $$
 
 \end{array}
 $$
 

-The authors then apply a dual based approach with Lagrange multiplier 
-to solve such a problem.
-They first introduce dual variables 
-$u_{hi}$, $v_{h}$, $\lambda_{i}$, and  $w_l$ for any 
-$h \in V$,$ i \in N$, and $l \in L$.
-They next replace the objective of reducing $\sum_{i \in N }q_i^2$
+
+They thus replace the objective of reducing
+$\sum_{i \in N }q_i^2$

 by the objective of reducing 
 by the objective of reducing 

-$$
+\begin{equation}

 \sum_{i \in N }q_i^2 + 
 \sum_{h \in V, l \in L } \delta.x_{hl}^2 
 + \sum_{h \in V }\delta.R_{h}^2 
 \sum_{i \in N }q_i^2 + 
 \sum_{h \in V, l \in L } \delta.x_{hl}^2 
 + \sum_{h \in V }\delta.R_{h}^2 

-$$
-where $\delta$ is a \JFC{ formalisation} factor.
-This allows indeed to get convex functions whose minimum value is unique.
+\label{eq:obj2}
+\end{equation}
+where $\delta$ is a regularisation factor.
+This indeed introduces quadratic fonctions on variables $x_{hl}$ and 
+$R_{h}$ and makes some of the functions strictly convex.
+
+The authors then apply a classical dual based approach with Lagrange multiplier 
+to solve such a problem~\cite{}.
+They first introduce dual variables 
+$u_{hi}$, $v_{h}$, $\lambda_{i}$, and  $w_l$ for any 
+$h \in V$,$ i \in N$, and $l \in L$.

 
 

-The proposed algorithm iteratively computes the following variables 
+\begin{equation}
+\begin{array}{l}
+L(R,x,P_{s},q,u,v,\lambda,w)=\\ 
+\sum_{i \in N} q_i^2 + q_i.  \left(
+\sum_{l \in L } a_{il}w_l^{(k)}-
+\lambda_iB_i
+\right)\\
++ \sum_{h \in V}
+v_h.\dfrac{\ln(\sigma^2/D_h)}{\gamma P_{sh} ^{2/3}} + \lambda_h P_{sh}\\
++ \sum_{h \in V} \sum_{l\in L}
+\left(
+\delta.x_{hl}^2  \right.\\
+\qquad \qquad + x_{hl}.
+\sum_{i \in N} \left( 
+\lambda_{i}.(c^s_l.a_{il}^{+} +
+c^r. a_{il}^{-} ) \right.\\
+\qquad \qquad\qquad \qquad +
+\left.\left. u_{hi} a_{il}
+\right)
+\right)\\
+ + \sum_{h \in V}
+\delta R_{h}^2 
+-v_h.R_{h} - \sum_{i \in N} u_{hi}\eta_{hi}
+\end{array}
+\end{equation}

 
 

+The proposed algorithm iteratively computes the following variables 
+untill the variation of the dual function is less than a given threshold.

 \begin{enumerate}
 \item $ u_{hi}^{(k+1)} = u_{hi}^{(k)} - \theta^{(k)}. \left(
  \eta_{hi}^{(k)} - \sum_{l \in L }a_{il}x_{hl}^{(k)} \right) $
 \begin{enumerate}
 \item $ u_{hi}^{(k+1)} = u_{hi}^{(k)} - \theta^{(k)}. \left(
  \eta_{hi}^{(k)} - \sum_{l \in L }a_{il}x_{hl}^{(k)} \right) $


@@ -108,10 +138,10 @@ The proposed algorithm iteratively computes the following variables
 $v_{h}^{(k+1)}= \max\left\{0,v_{h}^{(k)} -  \theta^{(k)}.\left( R_h^{(k)} - \dfrac{\ln(\sigma^2/D_h)}{\gamma.(P_{sh}^{(k)})^{2/3}}   \right)\right\}$
 \item 
   $\begin{array}{l}
 $v_{h}^{(k+1)}= \max\left\{0,v_{h}^{(k)} -  \theta^{(k)}.\left( R_h^{(k)} - \dfrac{\ln(\sigma^2/D_h)}{\gamma.(P_{sh}^{(k)})^{2/3}}   \right)\right\}$
 \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 -\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}
 $
 
 \end{array}
 $
 


@@ -125,14 +155,14 @@ $\theta^{(k)} = \omega / t^{1/2}$
  \item 
 $q_i^{(k)} = \arg\min_{q_i>0}
 \left(
  \item 
 $q_i^{(k)} = \arg\min_{q_i>0}
 \left(

-q^2 + q. 
+q_i^2 + q_i. 

 \left(
 \sum_{l \in L } a_{il}w_l^{(k)}-
 \lambda_i^{(k)}B_i
 \right)
 \right)$
 
 \left(
 \sum_{l \in L } a_{il}w_l^{(k)}-
 \lambda_i^{(k)}B_i
 \right)
 \right)$
 

-\item 
+\item \label{item:psh} 

 $
 P_{sh}^{(k)} 
 =
 $
 P_{sh}^{(k)} 
 =


@@ -171,4 +201,4 @@ c^r. a_{il}^{-} ) \right.\\
 $
 \end{enumerate}
 where the first four elements are dual variable and the last four ones are 
 $
 \end{enumerate}
 where the first four elements are dual variable and the last four ones are 

-primal ones  
\ No newline at end of file
+primal ones.
\ No newline at end of file