## 1 Introduction  
The demand for spectrum resources is increasing rapidly as wireless communication continues to develop. However, the limited spectrum resources severely restrict the further growth of communication capacity. Cognitive radio is an effective technology to enhance spectrum efficiency through the reasonable reuse of the authorized spectrum. The technology mainly includes three modes: spectrum interweave, spectrum overlay and spectrum underlay [1, 2]. Compared with interweave and overlay, underlay is simple to implement because no spectrum sensing is needed, and it has a better ability to realize spectrum sharing [3]. With underlay, in order to ensure a high priority of the primary user's (PU) access of the spectrum, the transmit power of the cognitive users must be kept under the interference tolerance threshold of the PU. Therefore, power control becomes a key issue in optimizing the overall performance of the cognitive network [4]. In [5], [PERSON] et al. investigate transmit power control for an underlay cognitive radio network by using a deep learning method that determines its own transmit power based solely on its local channel state information (CSI). In [6], [PERSON] et al. develop an optimal and novel joint antenna selection and power adaptation rule that minimizesthe average symbol error probability of a secondary user that is subject to two practically well-motivated constraints. [PERSON] et al. [7] propose two optimal power control schemes from the long-term and short-term perspectives for a cognitive low orbit satellite constellation with terrestrial networks, which aims to maximize the delay-limited capacity and minimize the outage probability, respectively. In [8], [PERSON] et al. propose a dynamic multiobjective approach for power and spectrum allocation in a cognitive-based environment and propose a dynamic resource allocation algorithm comprising a hybrid initialization method and feasible point generation mechanisms to solve the dynamic multiobjective optimization problem. In [9], two efficient and low-complexity power control strategies are proposed for an ambient backscatter-based spectrum-sharing network, and with the backscatter prominent, there is no need to estimate all users' CSI. The relay technique can expand the transmission distance and improve the transmission reliability of the system. Integrating relay and cognitive techniques can further improve the transmission performance of the system [10; 11]. In [12], the closed-form expression of outage probability for a cognitive multi-hop relay network is derived over Rayleigh fading channels, and an optimization problem to minimize the outage probability of the cognitive relay network is formulated and solved. In [13], a novel decentralized scheduling technique is developed for the cognitive multi-user multi-relay network, which operates on an incremental relaying mechanism and derives the outage probability of the secondary network for both the decode-and-forward (DF) and amplify-and-forward (AF) strategies. The transmission performance of a two-way AF cognitive relay network considering the influence of the primary network is studied in [14], and closed-forms of outage probability and bit error rate are derived. In [15], [PERSON] et al. propose a dynamic power transmission scheme for non-orthogonal multiple access (NOMA) cognitive relay networks and derive the closed-form expressions of outage probability and average sum rate. [PERSON] and [PERSON] investigate relay selection in a two-way full-duplex AF relay network [16] and derive the system outage probability and bit error rate. In [17], [PERSON] et al. investigate the energy efficiency and the spectral efficiency performance of multi-hop full duplex cognitive relay networks. Relaying often causes energy consumption issues, and forcing idle users to use their own energy to help the relay is difficult. Therefore, the energy issue in cognitive relay networks has become a topic of substantial research interest [18].