A superlinearly and globally convergent algorithm for power control and resource allocation with general interference functions

In wireless networks, users are typically coupled by interference. Hence, resource allocation can strongly depend on receive strategies, such as beamforming, CDMA receivers, etc. We study the problem of minimizing the total transmission power while maintaining individual quality-of-service (QoS) values for all users. This problem can be solved by the fixed-point iteration proposed by Yates (1995) as well as by a recently proposed matrix-based iteration (Schubert and Boche, 2007). It was observed by numerical simulations that the matrix-based iteration has interesting numerical properties, and achieves the global optimum in only a few steps. However, an analytical investigation of the convergence behavior has been an open problem so far. In this paper, we show that the matrix-based iteration can be reformulated as a Newton-type iteration of a convex function, which is not guaranteed to be continuously differentiable. Such a behavior can be caused by ambiguous representations of the interference functions, depending on the choice of the receive strategy. Nevertheless, superlinear convergence can be shown by exploiting the special structure of the problem. Namely, the function is convex, locally Lipschitz continuous, and an invertible directional derivative exists for all points of interest.