  A variant of variationally optimized perturbation, incorporating
renormalization group properties in a straightforward way, uniquely fixes the
variational mass interpolation in terms of the anomalous mass dimension. It is
used at three successive orders to calculate the nonperturbative ratio
$F_\pi/\Lambda$ of the pion decay constant and the basic QCD scale in the MSbar
scheme. We demonstrate the good stability and (empirical) convergence
properties of this modified perturbative series for this quantity, and provide
simple and generic cures to previous problems of the method, principally the
generally non-unique and non-real optimal solutions beyond lowest order. Using
the experimental $F_\pi$ input value we determine \Lambda^{n_f=2}\simeq
359^{+38}_{-25} \pm 5 MeV and \Lambda^{n_f=3}=317^{+14}_{-7} \pm 13 MeV, where
the first quoted errors are our estimate of theoretical uncertainties of the
method, which we consider conservative. The second uncertainties come from the
present uncertainties in F_\pi/F and F_\pi/F_0, where F (F_0) is $F_\pi$ in the
exact chiral SU(2) (SU(3)) limits. Combining the \Lambda^{n_f=3} results with a
standard perturbative evolution provides a new independent determination of the
strong coupling constant at various relevant scales, in particular \alpha_S
(m_Z) =0.1174 ^{+.0010}_{-.0005} \pm .001 \pm .0005_{evol} and
\alpha_S^{n_f=3}(m_\tau)= 0.308 ^{+.007}_{-.004} \pm .007 \pm .002_{evol}. A
less conservative interpretation of our prescriptions favors central values
closer to the upper limits of the first uncertainties. The theoretical accuracy
is well comparable to the most precise recent {\em single} determinations of
\alpha_S, including some very recent lattice simulation determinations with
fully dynamical quarks.
