How to write a semidefinite relaxation on a matrix rank constraint?

1

I have an optimization variable matrix \mathbf{X}=\mathbf{W}\mathbf{W}^{H} where \mathbf{W} \in \mathcal{C}^{N\times M} and \mathbf{W}^{H} denotes the conjugate transpose of \mathbf{W}. In order to write this in a convex form, a SDP relaxation needs to be used in replacement of the inherent rank constraint in \mathbf{X}; as \mathbf{X} >= \mathbf{W}\mathbf{W}^{H} or equivalently
\begin{bmatrix} \mathbf{X} & \mathbf{W}\\ \mathbf{W}^{H} & 1 \end{bmatrix}>=0.

My questioan is how to write this constraint in a disciplined convex programming framework using CVX ?

2

[ X, W ; W’, eye(M) ] == hermitian_semidefinite(M+N)

3

Thanks Michael!