Given a problem of the type minimize \;\; ||Ax - b||_1 where A is eg. \mathbb{R}_{15 \times 15}.

I want to impose constraints of the type:

\begin{bmatrix}
x_1 \\
x_2 \\
x_3
\end{bmatrix} =
\begin{bmatrix}
\alpha_1 \; x_6 \\
\alpha_1 \; x_7 \\
\alpha_1 \; x_8
\end{bmatrix}
= \begin{bmatrix}
\alpha_2 \; x_{11} \\
\alpha_2 \; x_{12} \\
\alpha_2 \; x_{13}
\end{bmatrix}

where the here mentioned x and \alpha are strictly positive.

Due to the x and \alpha being strictly positive, the constraints can be rewritten to a convex form using the convexifying operations from geometric programming (GP). But, CVX will not accept this due to the ‘no-product rule’

This can be circumvented by rewriting the whole problem on a geometric programming form.

I have two reasons for not doing this:

- With the iterative solver CVX uses for GP, the problems explode in terms of variables, and even relatively small problems cannot be handled.
- The rewriting to a GP form can be quite tedious and I can imagine problems where it is simply not possible.

Can I write this ratio constraint on a form that will be accepted by CVX without using GP?

Thank you for your time. I hope you can help.