X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/desynchronisation-controle.git/blobdiff_plain/120c13fc4faf264d8b1d3e94c4799860de9e3dea..e6cd9df3d469f7916d513610d3bc22ab055f790a:/IWCMC14/convexity.tex

diff --git a/IWCMC14/convexity.tex b/IWCMC14/convexity.tex
index a663986..0f7bf7e 100644
--- a/IWCMC14/convexity.tex
+++ b/IWCMC14/convexity.tex
@@ -10,7 +10,7 @@ v_h^{(k)}.\dfrac{\ln(\sigma^2/D_h)}{\gamma p ^{2/3}} + \lambda_h^{(k)}p
 $.
 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 
+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 
@@ -25,7 +25,7 @@ in equation~(\ref{eq:obj2}) by
 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.
+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.
 
@@ -38,10 +38,12 @@ $$
 \begin{array}{rcl}
 f'(p) &=& -2/3.v_h.\dfrac{\ln(\sigma^2/D_h)}{\gamma p^{5/3}} + \lambda_h + 
 8/3.\delta_p p^{5/3} \\
-&& \dfrac {8/3.\delta_p p^{10/3} + \lambda_h p^{5/3} -2/3.v_h\ln(\sigma^2/D_h)  }{p^{5/3}}
+& = & \dfrac {8/3\gamma.\delta_p p^{10/3} + \lambda_h p^{5/3} -2/3.v_h\ln(\sigma^2/D_h)  }{p^{5/3}}
 \end{array}
 $$
 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 $\lambda_h$. 
+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. 
 
   
\ No newline at end of file