Wireless network optimization by Perron-Frobenius theory

A basic question in wireless networking is how to optimize the wireless network resource allocation for utility maximization and interference management. In this paper, we present an overview of a Perron-Frobenius theoretic framework to overcome the notorious non-convexity barriers in wireless utility maximization problems. Through this approach, the optimal value and solution of the optimization problems can be analytically characterized by the spectral property of matrices induced by nonlinear positive mappings. It also provides a systematic way to derive distributed and fast-convergent algorithms and to evaluate the fairness of resource allocation. This approach can even solve several previously open problems in the wireless networking literature, e.g., Kandukuri and Boyd (TWC 2002), Wiesel, Eldar and Shamai (TSP 2006), Krishnan and Luss (WCNC 2011). More generally, this approach links fundamental results in nonnegative matrix theory and (linear and nonlinear) Perron-Frobenius theory with the solvability of non-convex problems. In particular, for seemingly nonconvex problems, e.g., max-min wireless fairness problems, it can solve them optimally; for truly nonconvex problems, e.g., sum rate maximization, it can even be used to identify polynomial-time solvable special cases or to enable convex relaxation for global optimization. To highlight the key aspects, we also present a short survey of our recent efforts in developing the nonlinear Perron-Frobenius theoretic framework to solve wireless network optimization problems with applications in MIMO wireless cellular, heterogeneous small-cell and cognitive radio networks. Key implications arising from these work along with several open issues are discussed.