We present a novel convex-optimization-based approach to the solutions of a family of problems involving constant modulus signals. The family of problems includes the constant modulus and the constrained constant modulus, as well as the modified constant modulus and the constrained modified constant modulus. These solutions are shown to constitute semidefinite programs (SDP), thus enabling efficient interior-point methods with polynomial time complexity. The performance  of the proposed solutions, demonstrated in several simulated experiments for the task of blind beamforming, is shown to be superior to existing methods.