Ratio constraints in CVX

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:

  1. With the iterative solver CVX uses for GP, the problems explode in terms of variables, and even relatively small problems cannot be handled.
  2. 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.

No. These constraints are non-convex. CVX cannot handle this. In fact, the objective function is not GP compatible either.