I have been trying to minimize $\mathrm{Tr}((G^TG)^{-1})$ using CVX. I have formulated it in the following SDP structure, using Schur Complement. Here is the formulation: $$\mathbf{minimise} \ \ t \\\mathbf{subject\ \ to}$$ $$\begin{bmatrix} I & G \\G^T & -X \end{bmatrix} \succeq 0 \qquad \begin{bmatrix} Z & I \\ I & X \end{bmatrix} \succeq 0 \qquad \mathop{\textrm{Tr}}(Z)\leq t$$
For the first matrix (the matrix with $G$ in it), the inequality inferred by Schur complement is $X \preceq -G^TG$ (this requires $I$ to be positive definite, which it is by definition)
For the other matrix, the inequality inferred by Schur complement is $Z \succeq X^{-1}$ (this requires $Z$ to be positive definite)
So, I did plug this in CVX but the result does not seem legitimate. Can someone please tell me if my formulation is correct?
Your model is wrong, which sort of is hinted by the obvious problem that the first constraint forces $X$ to be negative semidefinite, while the seconds forces it to be positive semidefinite
You have $Z \geq (G^TG)^{-1}$, and you then introduce a new variable $X$ to model $G^TG$. However, I don't see why you introduce the constraint $X \preceq -G^TG$. What you must have is $Z \succeq X^{-1} \succeq (G^TG)^{-1}$, which means $X \preceq G^TG$, a non-convex set which cannot be written as a linear semidefinite constraint.
A simple way to realize that this cannot be cast as a linear semidefinite program is the fact that the function basically is $1/x^2$ in the scalar case, easily seen to be non-convex.