We present a novel convex-optimization-based solution to blind linear multiuser detection in direct-sequence code division multiple access systems. The solution is based on a convex low-rank approximation of the linearly constrained constant modulus cost function, thus guaranteeing its global minimization. Further, it can be cast as a semidefinite program, implying that it can be solved using interior-point techniques with polynomial time complexity. The solution is parameter free and is shown to be superior to existing solutions in terms of output signal-to-interference-plus-noise ratio and bit error rate, especially for a small number of samples.