I still don’t understand what your problem is, to include what’s happening with the Schur Complement (that’s to deal with z’R’Rz ?) . This is your first mention of sum(b_j Q_j), b_j , but if b_j are known, as per your first post, then setting this equal to a known quantity is just an affine constraint, not quadratic (in the variables being optimized). Of course you can impose a semidefinite constraint on sum(b_j* Q_j) without use of Cholesky factorization, but then you would not have a connection to the previous (Schur Complement?) constraint.
I will say again, your problem must be convex in order to use CVX - proving it is convex is your responsibility… If it is not, you need another tool which can handle non-convexities, such as maybe PENLAB if there are semidefinite constraints. But again, I don’t understand your problem, and have not reached a determination whether there’s a convex formulation (exact, or perhaps relaxing some constraint(s)).
Please read and take to heart Why isn't CVX accepting my model? READ THIS FIRST! .