  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...
