  We consider a multi-user multiple-input single-output (MISO) communications
system which is assisted by a hybrid active-passive reconfigurable intelligent
surface (RIS). Unlike conventional passive RISs, hybrid RIS is equipped with a
few active elements with the ability to reflect and amplify incident signals to
significantly improve the system performance. Towards a fairness-oriented
design, we maximize the minimum rate among all users through jointly optimizing
the transmit beamforming vectors and RIS reflecting/amplifying coefficients.
Combining tools from block coordinate ascent and successive convex
approximation, the challenging nonconvex problem is efficiently solved by a
low-complexity iterative algorithm. The numerical results show that a hybrid
RIS with 4 active elements out of a total of 50 elements with a power budget of
-1 dBm offers an improvement of up to 80% to the considered system, while that
achieved by a fully passive RIS is only 27%.
