When both PHI and phi_1 are optimization variables in CVX, and PHI = phi_1 * phi_1’, how can this relationship be expressed in CVX?
That is a non-convex relationship (rank one constraint), so it can’t be expressed n CVX.
The semidefinite relaxation PHI \succeq phi* phi^T, whichrelaxes (ignores) the rank one constraint can be expressed in sdp mode as
[1 phi';phi PHI] >= 0
Is the modified expression [1 phi’;phi PHI] >= 0 equivalent to the previous expression PHI = phi * phi’?
No. It’s a senidefinite relaxation.