Optimality of Radio Power Control via Fast-Lipschitz Optimization
  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. 
Martin Jakobsson and Carlo Fischione
math.OC