# 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.

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

EDIT: Apologies if this is in the wrong category.

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