-
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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 $\lambda_i$. 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: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 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
+$$
+\begin{array}{rcl}
+f'(p) &=& -2/3.v_h.\dfrac{\ln(\sigma^2/D_h)}{\gamma p^{5/3}} + \lambda_h +
+8/3.\delta_p p^{5/3} \\
+&& \dfrac {8/3.\delta_p p^{10/3} + \lambda_h p^{5/3} -2/3.v_h\ln(\sigma^2/D_h) }{p^{5/3}}
+\end{array}
+$$
+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 $\lambda_h$.
+
+
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