- #1
perplexabot
Gold Member
- 328
- 5
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:
<br /> \begin{equation*}<br /> \begin{aligned}<br /> & \underset{G}{\text{minimize}}<br /> & & \mathrm{trace}((G^TG)^{-1}) \\<br /> \end{aligned}<br /> \end{equation*}<br />
which is equivalent to
<br /> \begin{equation*}<br /> \begin{aligned}<br /> & \underset{t, G}{\text{minimize}}<br /> & & \mathrm{t} \\<br /> & \text{subject to}<br /> && t \geq\mathrm{trace}((G^TG)^{-1})<br /> \end{aligned}<br /> \end{equation*}<br />
which is equivalent to
<br /> \begin{equation*}<br /> \begin{aligned}<br /> & \underset{t, G, X, Z}{\text{minimize}}<br /> & & \mathrm{t} \\<br /> & \text{subject to}<br /> && t \geq\mathrm{trace}(Z) \\<br /> &&&\begin{bmatrix} X & G^T \\G & I \end{bmatrix} \succeq 0 \qquad \\<br /> &&&\begin{bmatrix} Z & I \\ I & X \end{bmatrix} \succeq 0 \qquad<br /> \end{aligned}<br /> \end{equation*}<br />
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.
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:
<br /> \begin{equation*}<br /> \begin{aligned}<br /> & \underset{G}{\text{minimize}}<br /> & & \mathrm{trace}((G^TG)^{-1}) \\<br /> \end{aligned}<br /> \end{equation*}<br />
which is equivalent to
<br /> \begin{equation*}<br /> \begin{aligned}<br /> & \underset{t, G}{\text{minimize}}<br /> & & \mathrm{t} \\<br /> & \text{subject to}<br /> && t \geq\mathrm{trace}((G^TG)^{-1})<br /> \end{aligned}<br /> \end{equation*}<br />
which is equivalent to
<br /> \begin{equation*}<br /> \begin{aligned}<br /> & \underset{t, G, X, Z}{\text{minimize}}<br /> & & \mathrm{t} \\<br /> & \text{subject to}<br /> && t \geq\mathrm{trace}(Z) \\<br /> &&&\begin{bmatrix} X & G^T \\G & I \end{bmatrix} \succeq 0 \qquad \\<br /> &&&\begin{bmatrix} Z & I \\ I & X \end{bmatrix} \succeq 0 \qquad<br /> \end{aligned}<br /> \end{equation*}<br />
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: