# Changing to SDP (optimization)

1. Nov 25, 2015

### perplexabot

Hey all. Let me get right to it!

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

I used schur complement to do the following:
\begin{equation*} \begin{aligned} & \underset{G}{\text{minimize}} & & \mathrm{trace}((G^TG)^{-1}) \\ \end{aligned} \end{equation*}
which is equivalent to
\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*}
which is equivalent to
\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*}

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

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.

EDIT: Apologies if this is in the wrong category.

Last edited: Nov 25, 2015
2. Nov 30, 2015

### Greg Bernhardt

Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?

3. Dec 2, 2015