X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/desynchronisation-controle.git/blobdiff_plain/eff67eeac67ec967add0eb7e5e90d500ac32a491..1216fb04e58a581d9a4d2e77355aaf47bf989447:/IWCMC14/convexity.tex

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+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.
+
+  
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