Quasi-Convexity of the Asymptotic Channel MSE in Regularized Semi Blind Estimation

In this paper, the quasi-convexity of a sum of quadratic fractions in the form Σ_i=1ⁿ [(1+ci x²)/((1+dix)²)] is demonstrated where ci and di are strictly positive scalars, when defined on the positive real axis R⁺. It will be shown that this quasi-convexity guarantees it has a unique local (and hence global) minimum. Indeed, this problem arises when considering the optimization of the weighting coefficient in regularized semi-blind channel identification problem, and more generally, is of interest in other contexts where we combine two different estimation criteria. Note that V. Buchoux have noticed by simulations that the considered function has no local minima except its unique global minimum but this is the first time this result, as well as the quasi-convexity of the function is proved theoretically.