X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/desynchronisation-controle.git/blobdiff_plain/eff67eeac67ec967add0eb7e5e90d500ac32a491..ae93ee0d91e2a8f31ec45f7b77f9e81294630f07:/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 strictly convex if and only if 
+$\lambda_h$ is not null. This asymptotic configuration may arise due to 
+the definition of $\lambda_h$. Worth, in this case,  the function is 
+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 + 
+\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:obj2p}
+\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 separately study the influence of each factor.
+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
+$$
+\begin{array}{rcl}
+f'(p) &=& -2/3.v_h.\dfrac{\ln(\sigma^2/D_h)}{\gamma p^{5/3}} + \lambda_h + 
+8/3.\delta_p p^{5/3} \\
+& = & \dfrac {8/3.\delta_p p^{10/3} + \lambda_h p^{5/3} -2/3\gamma.v_h\ln(\sigma^2/D_h)  }{p^{5/3}}
+\end{array}
+$$
+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 $\lambda_h$ since the discriminant 
+is positive. 
+
+  
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