We consider the problem of sum-rate maximization in multiple-input multiple-output (MIMO) amplify-and-forward relay networks with multi-operator. The aim is to design the MIMO relay amplification matrix (i.e., the relay beamformer) to maximize the achievable communication sum rate through the relay. The design problem for the case of single-antenna users can be cast as a non-convex optimization problem, which, in general, belongs to a class of NP-hard problems. We devise a method based on the minorization-maximization technique to obtain quality solutions to the problem. Each iteration of the proposed method consists of solving a strictly convex unconstrained quadratic program. This task can be done quite efficiently, such that the suggested algorithm can handle the beamformer design for relays with up to similar to 70 antennas within a few minutes on an ordinary personal computer. Such a performance lays the ground for the proposed method to be employed in medium-scale (or lower regime massive) MIMO scenarios.