Wireless Max–Min Utility Fairness With General Monotonic Constraints by Perron–Frobenius Theory

This paper presents a systematic approach for solving wireless max–min utility fairness optimization problems in multiuser wireless networks with general monotonic constraints. These problems are often challenging to solve due to their nonconvexity. By establishing a connection between this class of optimization problems and the class of conditional eigenvalue problems that can be addressed by a generalized nonlinear Perron–Frobenius theory, we show how these problems can be solved optimally using an iterative algorithm that converges geometrically fast. The mathematical development in this paper unifies previous work and allows us to handle a broader class of competitive utility functions with general nonlinear monotonic constraints. Several representative applications illustrate the effectiveness of the proposed framework, including the max–min quality-of-service subject to robust interference temperature constraints in cognitive radio networks, the min–max weighted mean-square error subject to signal-to-interference-and-noise ratio constraints in multiuser downlink systems, the max–min throughput subject to nonlinear power constraints in energyefficient wireless networks, the max–min sigmoid utility in multimedia wireless networks, and the min–max outage probability subject to outage constraints in heterogeneous wireless networks. Numerical results are presented to demonstrate the fast-convergence behavior of the algorithms to the optimal fixed-point solution characterized by our generalized nonlinear Perron–Frobenius theoretic framework.