Abstract

In this paper, we investigate the joint design of transmit beamforming and power control to maximize the weighted sum rate in the multiple-input single-output (MISO) cognitive radio network constrained by arbitrary power budgets and interference temperatures. The nonnegativity of the physical quantities, e.g., channel parameters, powers, and rates, is exploited to enable key tools in nonnegative matrix theory, such as the (linear and nonlinear) Perron-Frobenius theory, quasi-invertibility, and Friedland-Karlin inequalities, to tackle this nonconvex problem. Under certain (quasi-invertibility) sufficient condition, we propose a tight convex relaxation technique that relaxes multiple constraints to bound the global optimal value in a systematic way. Then, a single-input multiple-output (SIMO)-MISO duality is established through a virtual dual SIMO network and Lagrange duality. This SIMO-MISO duality is equivalent to the zero Lagrange duality gap condition that connects the optimality conditions of the primal MISO network and the virtual dual SIMO network. Moreover, by exploiting the SIMO-MISO duality, an algorithm is developed to solve the sum rate maximization problem optimally. Numerical examples demonstrate the computational efficiency of our algorithm when the number of transmit antennas is large. Index Terms Optimization, convex relaxation, cognitive radio network, nonnegative matrix theory, quasi-invertibility, Karush-Kuhn-Tucker conditions, Perron-Frobenius theorem. I.