  Considering the spectrum sharing system (SSS) coexisting with multiple
primary networks, we have employed a well-designed reconfigurable intelligent
surface (RIS) to control the radio environments of wireless channels and
relieve the scarcity of the spectrum resource in this work. Specifically, the
enhancement of the spectral efficiency of the secondary user in the considered
SSS is decomposed into two subproblems which are a second-order cone
programming (SOCP) and a fractional programming of the convex quadratic form
(CQFP), respectively, to optimize alternatively the beamforming vector at the
secondary access point (S-AP) and the reflecting coefficients at the RIS. The
SOCP subproblem is shown as a concave problem, which can be solved optimally
using standard convex optimization tools. The CQFP subproblem can be solved by
a low-complexity method of gradient-based linearization with domain (GLD),
providing a sub-optimal solution for fast deployment. Taking the discrete phase
control at the RIS into account, a nearest point searching with penalty (NPSP)
method is also developed, realizing the discretization of the phase shifts of
the RIS in practice. The simulation results indicate that both GLD and NPSP can
achieve an excellent performance.
