beaucoup decourbes

[desynchronisation-controle.git] / IWCMC14 / convexity.tex

diff --git a/IWCMC14/convexity.tex b/IWCMC14/convexity.tex
index a663986869692dc80733037033e9241646508fe9..0f7bf7e7e6f076cf4a54ccf8c6c0db1d854c787f 100644 (file)
--- 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 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 
 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 $\delta$.  
 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.
 
 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} \\
 \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}
 $$
 which is positive if and only if the numerator is.
 \end{array}
 $$
 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. 

 
   
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