Impact of Interference Coupling - Loss of Convexity

In interference coupled wireless systems, where it is not possible to "orthogonalize" all the users in the system, we characterize the impact of interference coupling on the convexity of certain utility functions and problems. We introduce a general class of competitive user utility functions and natural competitive user utility functions. We further introduce the signal-to-interference based utility functions, which are based on physical layer parameters in wireless systems. We prove the conditions, which when satisfied result in a competitive user utility function being a signal-to-interference ratio based utility function. We further show that there exists no natural competitive user utility function, which is convex or concave. Furthermore, we show that a sum of weighted combination of natural competitive user utility functions is not convex or concave. Such functions are commonly encountered in wireless communication systems, e.g. rate or MMSE as a function of signal-to-interference ratio. We show that such rate maximization or MMSE minimization problems are not convex programs under our specified conditions.