Beamforming in multiple-input multiple-output (MIMO) systems should take interference mitigation into account. However, for beamform design, accurate channel state information (CSI) is needed, which is often difficult to obtain due to channel variability, feedback overhead, or hardware constraints. For example, amplify-and-forward (AF) relays passively forward signals without measurement, precluding full CSI acquisition to and from the relay. To address these issues, this paper introduces a novel prediction-assisted optimization (PAO) framework for beamform design in AF relay-assisted multiuser MIMO systems. The proposed solution in the AF relay aims at maximizing the signal-plus-interference-to-noise ratio (SINR). Unlike other methods, PAO relies solely on received power measurements, making it suitable for scenarios where CSI is unreliable or unavailable. PAO consists of two stages: a supervised-learning-based neural network (NN) that predicts the positions of transmitters using signal observations, and an optimization algorithm, guided by a digital twin (DT), that iteratively refines the beam direction of the relay in a simulated radio environment. As a key contribution, we validate the proposed framework using realistic measurements collected on a custom-built experimental millimeter wave (mmWave) platform, which enables training of the NN model under practical wireless conditions. The estimated information is then used to update the digital twin with knowledge of the surrounding environment, enabling online optimization. Numerical results show the trade-off between localization accuracy and beamforming performance and confirm that PAO maintains robustness even in the presence of localization errors while reducing the need for real-world measurements.