In this letter, we consider a network of single-antenna sensors that aim at the estimation of an unknown deterministic parameter. The sensors collect the observations and forward them to a fusion center via applying a phase-only beamforming weight. The derivation of the optimal beamforming weights requires the solution of a unit-modulus quadratic program (UQP), which in the relevant literature is solved via the semidefinite relaxation (SDR) technique or via a variation of the analytic constant modulus algorithm. The former achieves better performance, though it exhibits high computational cost that increases drastically with the number of sensors. The latter requires much less complexity, though it achieves worse performance. To that end, we propose an efficient algorithm for the solution of UQPs based on the alternating direction method of multipliers. The new approach achieves almost identical performance to that of the SDR-based approach while exhibiting significantly reduced computational complexity. The convergence of the proposed algorithm to a Karush-Kuhn-Tucker point is theoretically studied, and its effectiveness is verified via numerical results.