Consider the max-min fairness linear transceiver design for a multi-user MIMO interference channel. Assuming perfect channel knowledge, this problem can be formulated as the maximization of minimum SINR utility, subject to individual power constraints at each transmitter. In this paper, it is shown that when the number of antennas at each transmitter (receiver) is at least two and at each receiver (transmitter) is at least three, the problem of checking whether the given target SINR is feasible is strongly NP-hard. A cyclic coordinate ascent algorithm is also proposed for this design problem. Monotonicity and global convergence to KKT solution of the proposed algorithm are proved.