Optimal precoder design for weighted sum-rate maximization in multiple-input multiple-output interference networks is studied. For this well known non-convex optimization problem, convex approximations based on interference alignment are developed, for both single-beam and multi-beam cases. Precoder design methods that consist of two phases, an interference alignment phase and a post-alignment optimization phase, are proposed. The interference alignment solution is taken as the input to the post-alignment optimization phase. For post-alignment weighted sum-rate maximization, novel iterative distributed algorithms are proposed based on the developed convex approximations. Simulation results show that the proposed algorithms achieve promising weighted sum-rate gains over existing interference alignment algorithms. Interestingly, for the multi-beam case, significant gain is achieved at all SNRs, including the high SNR regime.