## 1 Introduction  
The anti-jamming problem has always been a hot topic in wireless communication domain. As a major threat to reliable communication, the jammer can launch noise-like signals to legitimate receiver to decrease communication quality. In order to cope with malicious jamming attacks, a lot of anti-jamming works have been studied that focused on power control [1; 2; 3; 4; 5; 6; 7; 8; 9; 10], frequency selection [11; 12; 13; 14; 15; 16; 17; 18; 19; 20], and spatial domain [21; 22; 23; 24; 25], and many effective anti-jamming technologies were proposed. In this paper, we investigate the anti-jamming hierarchical optimization problem in relay communication system. The power adjustment is considered as an effective method to respond to the jammer's attacks directly. In [1], the authors made receivers more robust with adaptive arrays utilizing power inversion algorithm in jamming environment. In [2], based on the competitive relationship between user and jammer, a Stackelberg game was proposed to model the anti-jamming communication hierarchical optimization problem in the presence of smart jammer. Considering it is hard for the user and jammer to obtain the accurate power and channel state information, the authors analyzed the anti-jamming performance with observation error in [3; 4; 5]. In [6], the authors studied the communication problem between in WBANs via Stackelberg frame facing jamming attacks. In [7], the authors investigated cross-layer anti-jamming joint optimization problem based on Q-learning algorithm. In [8], the authors investigated discrete power strategy optimization problem with imperfect information. In [9],the power allocation problem was studied in cognitive radio communication networks within different game frames. Taking the perspective of Stackelberg game, the authors combed the anti-jamming technologies under different scenarios in [10], moreover, an anti-jamming decision-making framework was also proposed based on the adversarial characteristics between user and jammer, incomplete information and so on. In the domain of relay communication [26; 27; 28; 29; 30; 31; 32; 33; 34; 35; 36; 37; 38; 39], the anti-jamming problem has also been widely investigated. In [27], the authors studied communication frequency selection optimization problem in unpredictable jamming environment. In [28], the authors formulated the relay selection and discrete power optimization problem under power constraint as a potential game, in which there existed more than one pure strategy Nash equilibrium (NE) and the optimal strategy profile was obtained. In [29], the cooperative relaying technology was investigated in vehicular networks. In [30], the authors studied the ongoing power control optimization problem via Stackelberg game and obtained the closed-form solutions of optimal power strategy. In [31], the authors decreased algorithm complexity and obtained the optimal joint strategy in a distributed wireless communication networks. In [32], the authors focused on game-theoretical approaches in cognitive communication and networking systems, in which a stochastic anti-jamming game was proposed to design the optimal adaptive defense strategies against cognitive malicious attackers. In [33], the authors designed an opportunistic wireless communication setting and modelled the asset selling problem as a game theoretic variant in the completely observable and the partially observable cases respectively. With the increasing complexity of communication scenarios and jamming modes, in [34; 35], the authors studied the optimal power allocation problem in friendly jamming and Gaussian noise jamming environments to increase secrecy sum rate in untrusted relay communication networks. In [36], the authors studied anti-jamming power optimization problem in multi-user relay communication system, a three-layer Stackelberg game was proposed and Stackelberg equilibrium (SE) was also obtained based on duality optimization theory.