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

The widely considered power constraints on optimizing power allocation in wireless networks, e.g., \(p_{n}\ge 0, \forall n\), and \(\sum _{n=1}^{N}{p_{n}}\le P_{\text {max}}\) where N and \(P_{\text {max}}\) are given constants, imply the constraints, i.e., \(p_{n}\le P_{\text {max}}, \forall n\). However, the related implicit constraints are regarded as redundant in the 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 that have slow convergence speeds. Using the water-filling problem as an illustration, we derive the structural properties of the optimal solutions based on Karush-Kuhn-Tucker conditions, propose a non-iterative closed-form optimal method, and use subgradient methods to solve the problem. Our theoretical analysis shows that the implicit constraints are not redundant, and their consideration can effectively speed up convergence of the used iterative methods and reduce the sensitivity to the chosen step sizes. Numerical results for the water-filling problem and another existing power allocation problem confirm the effectiveness of considering the implicit constraints.