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.
+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.
+Thus, the method follows its iterative calculus
+with an arbitrarely large value for $P_{sh}^{(k)}$. This leads to
+a convergence which is dramatically slow down.
+
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}
+\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
+In this equation we have first introduced new regularization 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.
+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.
+
+This proposed enhancement has been evaluated as follows:
+10 thresholds $t$, such that $1E-5 \le t \le 1E-3$, have
+been selected and for each of them,
+10 random configurations have been generated.
+For each one, we store the
+number of iterations which is sufficient to make the dual
+function variation smaller than this given threshold with
+the two approaches: either the original one ore the
+one which is convex guarantee.
+
+The Figure~\ref{Fig:convex} summarizes the average number of convergence
+iterations for each treshold value. As we can see, even if this new
+enhanced method introduces new calculus,
+it speeds up the algorithm and guarantees the convexity,
+and thus the convergence.
+\begin{figure*}
+\begin{center}
+\includegraphics[scale=0.5]{convex.png}
+\end{center}
+\caption{Original Vs Convex Guarantee Approaches}\label{Fig:convex}
+\end{figure*}
+
-
\ No newline at end of file
