Changing to SDP (optimization)

[perplexabot]

Hey all. Let me get right to it!

I have the following objective function: [tex]\mathbf{minimize} \ \ trace((G^TG)^{-1})[/tex]
I am trying to minimize it with CVX.

I used schur complement to do the following:
[tex]
\begin{equation*}
\begin{aligned}
& \underset{G}{\text{minimize}}
& & \mathrm{trace}((G^TG)^{-1}) \\
\end{aligned}
\end{equation*}
[/tex]
which is equivalent to
[tex]
\begin{equation*}
\begin{aligned}
& \underset{t, G}{\text{minimize}}
& & \mathrm{t} \\
& \text{subject to}
&& t \geq\mathrm{trace}((G^TG)^{-1})
\end{aligned}
\end{equation*}
[/tex]
which is equivalent to
[tex]
\begin{equation*}
\begin{aligned}
& \underset{t, G, X, Z}{\text{minimize}}
& & \mathrm{t} \\
& \text{subject to}
&& t \geq\mathrm{trace}(Z) \\
&&&\begin{bmatrix} X & G^T \\G & I \end{bmatrix} \succeq 0 \qquad \\
&&&\begin{bmatrix} Z & I \\ I & X \end{bmatrix} \succeq 0 \qquad
\end{aligned}
\end{equation*}
[/tex]

Those two matrices introduced by schur complement achieve the following two inequalities: [tex]X \geq G^TG[/tex] and [tex]Z \geq X^{-1}[/tex]

My question is, is this formulation correct?

Here are some links that may be worth the read if you are interested:
The work I did is based on the following similar example.
I have had some help at the official cvx forums.

Thank you for reading : ) Any comments, pointers or advice is much appreciated!

EDIT: Apologies if this is in the wrong category.

 

[perplexabot]

Answer here: http://math.stackexchange.com/questions/1555273/minimize-mathrmtrgtg-1
More discussion here: http://ask.cvxr.com/t/minimizing-trace-inverse-of-grammian/3135/35

Thank you for reading : )