Optimal Algorithms in Wireless Utility Maximization: Proportional Fairness Decomposition and Nonlinear Perron-Frobenius Theory Framework

We study the network utility maximization problems in wireless networks for service differentiation that optimize the Signal-to-Interference-plus-Noise Radio (SINR) and reliability under Rayleigh fading. Though seemingly nonconvex, we show that these problems can be decomposed into an optimization framework where each user calculates a payment for a given resource allocation, and the network uses the payment to optimize the performance of the user. We study three important examples of this utility maximization, namely the weighted sum logarithmic SINR maximization, the weighted sum inverse SINR minimization and the weighted sum logarithmic reliability maximization. These problems have hitherto been solved suboptimally in the literature. By exploiting the positivity, quasi-concavity and homogeneity properties in these problems and using the nonlinear Perron-Frobenius theory, we propose fixed-point algorithms that converge geometrically fast to the globally optimal solution. Numerical evaluations show that our algorithms are stable (free of parameter configuration) and computationally fast.