## Abstract

In this chapter, 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, a tight convex relaxation technique can relax 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 proved to have the zero duality gap that connects the optimality conditions of the primal MISO network and the virtual dual SIMO network. By exploiting the SIMO-MISO duality, we present an algorithm to optimally solve the sum rate maximization problem.

Original language | English |
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Title of host publication | Cognitive Radio Networks |

Subtitle of host publication | Performance, Applications and Technology |

Publisher | Nova Science Publisher Inc. |

Pages | 153-183 |

Number of pages | 31 |

ISBN (Electronic) | 9781536130690 |

ISBN (Print) | 9781536130683 |

State | Published - 01 01 2018 |

### Bibliographical note

Publisher Copyright:© 2018 Nova Science Publishers, Inc.

## Keywords

- Cognitive radio network
- Convex relaxation
- Karush-kuhn-tucker conditions
- Nonnegative matrix theory
- Optimization
- Perron-frobenius theorem
- Quasi-invertibility