It’s no more convex over here than it was at CVX variable U(N,N). How to make rank(U)=1? - #2 by Mark_L_Stone .
Please don’t start multiple topics on the same matter. And before posting again, please CAREFULLY read the link in my post there.
As for rank one constraints, sometimes they are relaxed. If the relaxed problem does not satisfy the rank one constraint, then you have not solved the original problem, and only get a bound on the optimal objective value of the original problem (lower bound for minimization problem).