 of achieving certain QoS requirements carries over to the problem of achieving SIR targets Vk > 0, k. The problem of achieving SIR targets , YK with minimum total transmission power was addressed by Yates in Yeung et al. [55]. The problem can be written as (7.10) where P C RR is a compact set of possible power vectors. The set P depends on the power constraints of the system under consideration. Typically, each user has a max- imum power limit Pk that must not be exceeded (i.e., P = {p: 0 However, other types of power constraints are possible (e.g., a sum-power con- straint, which is typical for broadcast transmission). Note that the problem (7.10) can be infeasible if the targets Y1,..., YK cannot be achieved. We need to assume feasibility. Also, the interference function I(p) must be strictly monotonic, with respect to the noise component, in addition to the prop- erties A1-A3. Consider the general power set (7.11) The presence of a constant noise component is important to ensure that the infimum (7.10) is actually attained. In this case, it follows from Yates [55] (see also the alternative proof in Schubert/Boche [51]) that problem (7.10) is solved by the following fixed-point iteration: Ak E K. (7.12) In each iteration, the users update their transmission powers depending on the observed interference power. This iteration converges component-wise to the unique global optimum of problem (7.10), regardless of the chosen initialization p 0 E P. This interference framework is quite basic, and it holds (sometimes in approxi- mation) for many interference scenarios. Despite its simplicity, a globally convergent C 2009 by Taylor & Francis Group, LLC Long Term Evolution: 3GPP LTE Radio and Cellular Technology algorithm (7.12), with a nice analytical structure and a known convergence behav- ior, exists [40,46]. 7.2.2.3 Exploiting Convexity The fixed-point iteration (7.12) has the advantage of being applicable to general interference functions. However, a better convergence behavior can be expected when convexity or c