In solving a robust version of regularized least squares with weighting, a certain scalar-valued optimization problem is required in order to determine the regularized robust solution and the corresponding robustified weighting parameters. This letter establishes that the required optimization problem does not have local, non-global minima over the interval of interest. This property is proved by resorting to a useful Schur complementation argument. The result is reassuring in that it demonstrates that the robust design procedure is well defined and that its optimal global solution can be determined without concerns about local minima.