  This paper considers a $K$-user multiple-input single-output (MISO)
interference channels for simultaneous wireless information and power transfer
(SWIPT), where each multi-antenna transmitter serves a single-antenna receiver
per user pair. All receivers perform simultaneously information processing and
energy harvesting (EH) based on the receive power-splitting (PS) architectures.
Assuming imperfect channel state information (CSI) at the transmitters, we
develop an optimal robust transceiver design scheme that minimizes the total
transmission power under the worst-case signal-to-interference-plus-noise ratio
(SINR) and energy harvesting (EH) constraints at the receivers, by jointly
optimizing transmit beamforming and receive PS ratio per receiver. When the CSI
uncertainties are bounded by ellipsoidal regions, it is shown that the
worst-case SINR and EH constraints per receiver can be recast into quadratic
matrix inequality forms. Leveraging semidefinite relaxation technique, the
intended robust beamforming and PS (BFPS) problem can be relaxed as a tractable
(centralized) semidefinite program (SDP). More importantly, relying on the
state-of-the-art alternating direction method of multipliers (ADMM) in convex
optimization, we propose a {\em decentralized} algorithm capable of computing
the optimal robust BFPS scheme with local CSI and limited information exchange
among the transmitters. It is shown the proposed decentralized algorithm is
guaranteed to converge to the optimal centralized solution. Numerical results
are provided to demonstrate the merits of the proposed approaches.
