Maximizing the sum rates in a multiuser Gaussian channel by power control is a nonconvex NP-hard problem that findes engineering application in code division multiple access (CDMA) wireless communication network. We study the problem of total data throughput maximization using power control in a CDMA network, where interference is a major source of performance impairment. Due to the broadcast nature of the wireless medium, the data rates in a wireless network are affected by interference when all the users transmit simultaneously over the same frequency band using CDMA. Power control is used to mitigate the effect of multiuser interference on performance and maximize the total data rates of all the users. The CDMA wireless network can be modeled by an information-theoretic interference channel that treats multiuser interference as additive Gaussian noise. A widely studied problem is to find the optimal power allocation that maximizes the sum rates over this multiuser Guassian channel, and this requires solving a nonconvex problem. This nonconvex problem also findes applications in the throughput maximization for digital subscriber line (DSL) wireline systems. The complexity of an exhaustive search is prohibitively expensive, since this optimization problem is NP-hard, and may even be hard to approximate. Chiang and Tan formulated the problem as a signomial program, and used a successive convex approximation method on geometric programming. An often used technique to tackle nonconvexity is the standard Lagrange dual relaxation of the problem in the power domain. However, the shortcoming of this approach is that there can exist a positive duality gap between the global optimal primal and optimal dual value of the main problem. Also, finding an optimal primal solution given an optimal dual solution, or finding an optimal dual solution given an optimal primal solution, is in general difficult. We adopt a reformulation-relaxation approach to tackle with the nonconvex power control optimization problem. Our reformulation possesses certain desirable properties, which enable the application of nonnegative matrix theory, especially Friedland-Karlin inequalities, to find the global optimal solution and motivate efficient relaxation techniques. In particular, we utilize the problem structure to develope suitablefast computational procedures for solving and computing useful bounds to the sum rate mximization problem. Furthermore, analytical solution to both the sum rate maximization problem and its relaxed problem can also be characterized by the spectra of specially crafted nonnegative matrices. So in this thesis , we extend and apply several fundamental nonnegative matrix inequalties 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 epsilon -suboptimal ity to compute the optimal power control allocation.A byproduct of our analysis is a refinement of the Friedland-Karlin inequalities and its application to an inverse problem in nonnegative matrix theory.