Advantage sdp relaxation?

what is the advantage of using [I x;xT z]>0 constraint rather than norm(x).^2<z constraint in cvx? In my problem the objective function is not linear and so I cannot use SDP.

I am afraid I don’t understand what you are asking. I would certainly NOT use the SDP form, but rather sum_square(x) ``<``= z. CVX 3.0 can handle norm(x).^2 <= z natively as well.