Optimizing power allocation in wireless networks: Are the implicit constraints really redundant?

Abstract

The widely considered power constraints on optimizing power allocation in wireless networks, e.g., non-negative individual power and limited sum of all the individual power, imply the constraints where each individual power is not greater than the limited sum. However, the related implicit constraints are generally regarded as redundant for algorithm design in most current studies. In this paper, we explore the question “Are the implicit constraints really redundant?” in the optimization of power allocation especially when using iterative methods (e.g., subgradient method) that have slow convergence speeds. Using the water-filling problem as an illustration, we first derive the structural properties of the optimal solutions based on Karush–Kuhn–Tucker conditions. Then we propose a non-iterative closed-form optimal method and use iterative methods (i.e., bisection method and subgradient method) to solve the problem. Our theoretical analysis shows that the implicit constraints are not redundant, and particularly, their consideration can effectively speed up the convergence of the subgradient method and reduce its sensitivity to the chosen step size. Numerical results for the water-filling problem and another existing power allocation problem demonstrate the effectiveness of considering the implicit constraints.