$v_{h}^{(k+1)}= \max\left\{0,v_{h}^{(k)} - \theta^{(k)}.\left( R_h^{(k)} - \dfrac{\ln(\sigma^2/D_h)}{\gamma.(P_{sh}^{(k)})^{2/3}} \right)\right\}$
\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 - \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}
$
\right)
\right)$
-\item
+\item \label{item:psh}
$
P_{sh}^{(k)}
=
-
\ No newline at end of file
+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.
+
+
\ No newline at end of file
