This study examines the effect of peak-to-average power ratio (PAPR) constraint on the transmit beamformer design problem with the goal of establishing a trade-off between the power efficiency (maximizing the average transmitted power) and other metrics such as the power level fluctuation in mainlobe, peak-sidelobe level (PSL), etc. Typically, unimodular weights are utilized in transmit beamforming to maximize the average transmitted power. Yet, unimodular weights maximize the power efficiency at the expense of other performance metrics. It is shown that even a slight relaxation of the design problem from the unimodular condition (PAPR = 1), say setting PAPR <= 1.1, results in a significant improvement in other performance metrics at a negligible loss of power efficiency. To achieve the trade-off between the metrics, an alternating direction method of multipliers (ADMM) based solution to the transmit beamformer design is given. The suggested solution is applicable to both narrowband and wideband beamformers and also to some other related problems such as the unimodular radar waveform design (code design) problem. (C) 2020 Elsevier Inc. All rights reserved.