In this paper, an efficient approach for weighted sum rate maximization (WSRMax) for zero-forcing (ZF) methods with general linear transmit covariance constraints (LTCCs) is proposed. This problem has been extensively studied separately for some special cases such as for sum power or per-antenna power constraint (PAPC). Due to some practical and regulatory requirements, these power constraints alone are not in general sufficient, which motivates the consideration of general LTCCs. On the other hand, the zero-forcing (ZF) is a simple linear precoding technique to mitigate inter-user interference. The problem of WSRMax for ZF methods with LTCCs was studied previously using a gradient descent algorithm with barrier functions, but this method was also shown to converge slowly. To derive an efficient solution to this problem, we first reformulate it as an equivalent minimax problem using Lagrangian duality. The obtained result in fact resembles BC-MAC duality but is specialized for ZF methods. We then combine alternating optimization and concave-convex procedure to efficiently compute a saddle point of the minimax problem. The proposed method is numerically shown to converge very fast and its complexity scales linearly with the number of users.