As a promising solution to improve communication quality, unmanned aerial vehicle (UAV) has been widely integrated into wireless networks. In this paper, for the sake of enhancing the message exchange rate between User1 (U1) and User2 (U2), an intelligent reflective surface (IRS)-and-UAV- assisted two-way amplify-and-forward (AF) relay wireless system is proposed, where U1 and U2 can communicate each other via a UAV-mounted IRS and an AF relay. Besides, an optimization problem of maximizing minimum rate is casted, where the variables, namely AF relay beamforming matrix and IRS phase shifts of two time slots, need to be optimized. To achieve a maximum rate, a low-complexity alternately iterative (AI) scheme based on zero forcing and successive convex approximation (LC-ZF-SCA) algorithm is put forward, where the expression of AF relay beamforming matrix can be derived in semi-closed form by ZF method, and IRS phase shift vectors of two time slots can be respectively optimized by utilizing SCA algorithm. To obtain a significant rate enhancement, a high-performance AI method based on one step, semidefinite programming and penalty SCA (ONS-SDP-PSCA) is proposed, where the beamforming matrix at AF relay can be firstly solved by singular value decomposition and ONS method, IRS phase shift matrices of two time slots are optimized by SDP and PSCA algorithms. Simulation results present that the rate performance of the proposed LC-ZF-SCA and ONS-SDP-PSCA methods surpass those of random phase and only AF relay. In particular, when total transmit power is equal to 30dBm, the proposed two methods can harvest more than 68.5% rate gain compared to random phase and only AF relay. Meanwhile, the rate performance of ONS-SDP-PSCA method at cost of extremely high complexity is superior to that of LC-ZF-SCA method.

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