  Users in millimeter-wave (mmWave) systems often exhibit diverse channel
strengths, which can negatively impact user fairness in resource allocation.
Moreover, exact channel state information (CSI) may not be available at the
transmitter, rendering suboptimal resource allocation. In this paper, we
address these issues within the context of energy efficiency maximization in
RIS-assisted mmWave systems. We first derive a tractable lower bound on the
achievable sum rate, taking into account CSI errors. Subsequently, we formulate
the optimization problem, targeting maximizing the system energy efficiency
while maintaining a minimum Jain's fairness index controlled by a tunable
design parameter. The optimization problem is very challenging due to the
coupling of the optimization variables in the objective function and the
fairness constraint, as well as the existence of non-convex equality and
fractional constraints. To solve the optimization problem, we employ the
penalty dual decomposition method, together with a projected gradient ascent
based alternating optimization procedure. Simulation results demonstrate that
the proposed algorithm can achieve an optimal energy efficiency for a
prescribed Jain's fairness index. In addition, adjusting the fairness design
parameter can yield a favorable trade-off between energy efficiency and user
fairness compared to methods that exclusively focus on optimizing one of these
metrics.
