Kuhn-tucker conditions for nuclear norm optimization methods for interference alignment

For designing optimized beamforming and zero forcing matrices for interference alignment, the usual way is to minimize the leakage interference. From the feasibility conditions of interference alignment, we have established equivalent conditions involving ranks of the interference and desired signal matrices and have proposed optimization of beamforming and zero-forcing matrices using ranks. Since minimization of rank of a matrix is a non-convex problem, it is approximated by a convex envelop of the rank which is the nuclear norm of the matrix. To maximize sum degrees of freedom for K users, we minimize the nuclear norm of sum of all interferences or simply minimizing the maximum nuclear norm of all interference matrices from all users. The nuclear norm of the desired signal matrix is constrained to be equal to the desired degrees of freedom, which guarantees a full rank matrix. We have developed the necessary and sufficient Kuhn-Tucker conditions for the optimization problems. We have also shown the convexity of our proposed optimization problems.