In wireless network resource allocation, the radio power control problems are often solved by fixed point algorithms. Although these algorithms give feasible problem solutions, such 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 solutions. The optimality is established by the recently proposed fast-Lipschitz optimization framework. To apply such a framework, the analysis is performed by a logarithmic transformation of variables that gives tractable fast-Lipschitz problems. It is shown how the logarithmic problem constraints are contractive by the standard or type-II standard assumptions on the power control problem, and how sets of cost functions fulfill the fast-Lipschitz qualifying conditions. The analysis on nonmonotonic interference function allows establishing 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 subhomogeneous power control problems. Given the generality of fast-Lipschitz optimization compared to traditional methods for resource allocation, it is concluded that such an optimization may help to determine the optimality of many resource allocation problems in wireless networks.