Nonnegative Matrix Inequalities and their Application to Nonconvex Power Control Optimization

Abstract

Maximizing the sum rates in a multiuser Gaussian channel by power control is a nonconvex NP-hard problem that finds engineering application in code division multiple access (CDMA) wireless communication network. In this paper, we extend and apply several fundamental nonnegative matrix inequalities initiated by Friedland and Karlin in a 1975 paper to solve this nonconvex power control optimization problem. Leveraging tools such as the Perron–Frobenius theorem in nonnegative matrix theory, we (1) show that this problem in the power domain can be reformulated as an equivalent convex maximization problem over a closed unbounded convex set in the logarithmic signal-to-interference-noise ratio domain, (2) propose two relaxation techniques that utilize the reformulation problem structure and convexification by Lagrange dual relaxation to compute progressively tight bounds, and (3) propose a global optimization algorithm with ϵ-suboptimality to compute the optimal power control allocation. A byproduct of our analysis is the application of Friedland–Karlin inequalities to inverse problems in nonnegative matrix theory.

Additional Information

© 2011 Society for Industrial and Applied Mathematics. Received by the editors April 28, 2009; accepted for publication (in revised form) by Y. C. Eldar April 26, 2011; published electronically September 29, 2011. This research has been supported in part by ARO MURI award W911NF-08-1-0233, NSF NetSE grant CNS-0911041, City University Hong Kong project grant 7200183, 7008087, and a grant from the American Institute of Mathematics.

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