  The $\alpha$-fair resource allocation problem has received remarkable
attention and has been studied in numerous application fields. Several
algorithms have been proposed in the context of $\alpha$-fair resource sharing
to distributively compute its value. However, little work has been done on its
structural properties. In this work, we present a lower bound for the optimal
solution of the weighted $\alpha$-fair resource allocation problem and compare
it with existing propositions in the literature. Our derivations rely on a
localization property verified by optimization problems with separable
objective that permit one to better exploit their local structures. We give a
local version of the well-known midpoint domination axiom used to axiomatically
build the Nash Bargaining Solution (or proportionally fair resource allocation
problem). Moreover, we show how our lower bound can improve the performances of
a distributed algorithm based on the Alternating Directions Method of
Multipliers (ADMM). The evaluation of the algorithm shows that our lower bound
can considerably reduce its convergence time up to two orders of magnitude
compared to when the bound is not used at all or is simply looser.
