This paper considers an intelligent reflecting sur-face (IRS)-aided simultaneous wireless information and power transfer (SWIPT) network, where multiple users decode data and harvest energy from the transmitted signal of a transmit-ter. The proposed design framework exploits the cost-effective IRS to establish favorable communication environment to improve the fair energy efficient. In particular, we study the max-min energy efficiency (EE) of the system by jointly designing the transmit information and energy beamforming at the base station (BS), phase shifts at the IRS, as well as the power splitting (PS) ratio at all users subject to the minimum rate, minimum harvested energy, and transmit power constraints. The formulated problem is non-convex and thus challenging to be solved. We propose two algorithms namely penalty-based and inner approximation (IA)-based to handle the non-convexity of the optimization problem. As such, we divide the original problem into two sub-problems and apply the alternating optimization (AO) algorithm for both proposed algorithms to handle it iteratively. In particular, in the penalty-based algorithm for the first sub-problem, the semi-definite relaxation (SDR) technique, difference of convex functions (DC) programming, majorization-minimization (MM) approach, and fractional programming theory are exploited to transform the non-convex optimization problem into a convex form that can be addressed efficiently. For the second sub-problem, a penalty-based approach is proposed to handle the optimization on the phase shifts introduced by the IRS with the proposed algorithms. For the IA-based method, we optimize jointly beamforming vectors and phase shifts while the PS ratio is solved optimally in the first sub-problem...
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