  Fixed point algorithms play an important role to compute feasible solutions
to the radio power control problems in wireless networks. Although these
algorithms are shown to converge to the fixed points that give feasible problem
solutions, the solutions often lack notion of problem optimality. This paper
reconsiders well known fixed point algorithms such as those with standard and
type-II standard interference functions, and investigates the conditions under
which they give optimal power control solutions by the recently proposed
Fast-Lipschitz optimization framework. When the qualifying conditions of
Fast-Lipschitz optimization apply, it is established that the fixed points are
the optimal solutions of radio power optimization problems. The analysis is
performed by a logarithmic transformation of variables that gives problems
treatable within the Fast-Lipschitz framework. It is shown how the logarithmic
problem constraints are contractive by the standard or type-II standard
assumptions on the original power control problem, and how a set of cost
functions fulfill the Fast-Lipschitz qualifying conditions. The analysis on non
monotonic interference function allows to establish a new qualifying condition
for Fast-Lipschitz optimization. The results are illustrated by considering
power control problems with standard interference function, problems with
type-II standard interference functions, and a case of sub-homogeneous power
control problems. It is concluded that Fast-Lipschitz optimization may play an
important role in many resource allocation problems in wireless networks.
