This paper deals with the design of a hybrid precoder for millimeter-wave MIMO systems. For the sake of concreteness, we consider an analog processing stage composed of a switching network with analog combining. The main contribution of this work consists on the proposal and evaluation of an optimization procedure based on a smart relaxation. The optimal hybrid precoder under a transmit power constraint is derived, after which, the analog precoding matrix is binarized. After an intuitive reasoning, we note that multiple solutions exist. Nevertheless, the (very) reduced computational complexity of the proposed optimization scheme makes it feasible for realistic implementations. Numerical results are reported to assess the performance of proposed hybrid precoder design.