  The cognitive radio (CR) network is a promising network architecture that
meets the requirement of enhancing scarce radio spectrum utilization.
Meanwhile, reconfigurable intelligent surfaces (RIS) is a promising solution to
enhance the energy and spectrum efficiency of wireless networks by properly
altering the signal propagation via tuning a large number of passive reflecting
units. In this paper, we investigate the downlink transmit power minimization
problem for the RIS-enhanced single-cell cognitive radio (CR) network
coexisting with a single-cell primary radio (PR) network by jointly optimizing
the transmit beamformers at the secondary user (SU) transmitter and the phase
shift matrix at the RIS. The investigated problem is a highly intractable due
to the coupled optimization variables and unit modulus constraint, for which an
alternative minimization framework is presented. Furthermore, a novel
difference-of-convex (DC) algorithm is developed to solve the resulting
non-convex quadratic program by lifting it into a low-rank matrix optimization
problem. We then represent non-convex rank-one constraint as a DC function by
exploiting the difference between trace norm and spectral norm. The simulation
results validate that our proposed algorithm outperforms the existing
state-of-the-art methods.
