Concave and Convex Interference Functions - General Characterizations and Applications

Many resource allocation problems can be studied within the framework of interference functions. Basic properties of interference functions are non-negativity, scale-invariance, and monotonicity. In this paper, we study interference functions with additional properties, namely convexity, concavity, and log-convexity. Such interference functions occur naturally in various contexts, e.g., adaptive receive strategies, robust power control, or resource allocation over convex utility sets. We show that every convex (resp. concave) interference function can be expressed as a maximum (resp. minimum) over a weighted sum of its arguments. This analytical insight provides a link between the axiomatic interference framework and conventional interference models that are based on the definition of a coupling matrix. We show how the results can be used to derive best-possible convex/concave approximations for general interference functions. The results have further application in the context of feasible sets of multiuser systems. Convex approximations for general feasible sets are derived. Finally, we show how convexity can be exploited to solve the problem of signal-to-interference-plus-noise ratio (SINR)-constrained power minimization with super-linear convergence.