The Reaearch Of Some Non-Convex Quadratic Fraction Optimization With Quadratic Constraints And The Application In Cognitive Radio Network

Quadratic optimization problems play a significant role in theory of mathematical programming. And, it is used widely in many important fields, such as production planning and management, communication engineering, network security, financial engineering, telecommunications system, speech recognition and so on. Therefore, the research on quadratic optimization is very significant.Cognitive radio is believed as a focus in the recent research. Cognitive radio allows secondary users (SUs) to share the allocated spectrum with primary users (PUs) under some conditions, which can improve spectrum utilization. The network can transmit signals efficiently in SUs and guarantee the quality of service (QoS) of the PUs. In the secondary network, we choose the help of relays, which can compensate for the effects of signal fading and shadowing.In this paper, we mainly discuss non-convex quadratic fraction optimization problems with two quadratic constraints, three quadratic constraints or more quadratic constraints respectively. Moreover, we deduce optimization models in cognitive relay network and give algorithms.The main contents are list as follows:(1) We discuss non-convex quadratic fraction optimization problems with two quadratic constraints, three quadratic constraints or more quadratic constraints respectively in this paper. First, we transform the fractional object function into quadratic function equivalently according to the equivalent statements. Second, one can obtain a8-approximation global optimum. In the algorithm, there are two key questions. The first is how to obtain upper and lower bounds.And the second is how to solve the sub-problem (QCQP) in inner loop. For the first question, we give two methods that be introduced in the body of the paper. But, the upper and lower bounds are only approximate. For the practical problems, we can give the upper or lower bound roughly by the prior knowledge of questions. For the second question, we advise different SDP method according to the number of constraints. When the number of constraints is2, the sub-problem (QCQP) can obtain exact solution by two methods. When the number of constraints is3, the sub-problem can obtain exact solution by the method that is shown in the paper. But, when the number of constraints is more than3, we can not find the exact solution of sub-problem. So, a randomized algorithm is designed to this question by related references.(2) Due to the rapid development of wireless communications, the cognitive radio is becoming crowded. In guaranteeing the QoS of the PUs, that is, the interference for PUs is less than a certain predefined threshold, the paper analyzes the performance of the SN in the cognitive relay network. Considers three system models respectively, the first is no communication between the primary transmitter (PT) and the secondary destination (SD), the secondary transmitter (ST) and the primary destination (PD). The second is opposite, that is, there is communication between them. Relays of the third system model use two-way communication, but relays the first two are both one-way communication. We design optimization model in the three systems respectively, including the SNR maximization model of the secondary destination (SD) and the globe transmit power minimization model of relays. The iterative algorithms are proposed to get the beamforming vector.(3) We deduce some optimization models in cognitive radio two-way relay network, and lay the foundation for the later research work.