On proportional fairness in nonconvex wireless systems

We investigate the concept of proportional fairness in the context of nonconvex rate regions. For convex rate regions, Kelly's definition of proportional fairness is equivalent to maximizing a sum of logarithms. For nonconvex rate regions, this equivalence does not always hold. In particular, a proportionally fair point in the sense of Kelly's definition may not exist. We propose the concept of local proportional fairness (LPF) and characterize its properties. Under mild conditions, an LPF point always exists, and all LPF points are Pareto optimal. Moreover, we show that in a multiantenna downlink with linear precoding, computing an LPF point represents a practically solvable problem. I.