We present novel convex-optimization-based solutions to the problem of blind beamforming of constant modulus signals, and to the related problem of linearly constrained blind beamforming of constant modulus signals. These solutions are based on a low-rank approximation, ensure global optimality and are parameter free, namely, do not contain any tuneable parameters and do not require any a-priori parameter settings. The proposed approach outperforms state-of-the-art both in terms of the number of required samples for convergence, and in terms of the beamformer output SINR.