  In this paper, we propose beamforming schemes to simultaneously transmit data
securely to multiple information receivers (IRs) while transferring power
wirelessly to multiple energy-harvesting receivers (ERs). Taking into account
the imperfection of the instantaneous channel state information (CSI), we
introduce a chance-constrained optimization problem to minimize the total
transmit power while guaranteeing data transmission reliability, data
transmission security, and power transfer reliability. As the proposed
optimization problem is non-convex due to the chance constraints, we propose
two robust reformulations of the original problem based on
safe-convex-approximation techniques. Subsequently, applying semidefinite
programming relaxation (SDR), the derived robust reformulations can be
effectively solved by standard convex optimization packages. We show that the
adopted SDR is tight and thus the globally optimal solutions of the
reformulated problems can be recovered. Simulation results confirm the
superiority of the proposed methods in guaranteeing transmission security
compared to a baseline scheme. Furthermore, the performance of proposed methods
can closely follow that of a benchmark scheme where perfect CSI is available
for resource allocation.
