How can I model this Hyperbolic constraint?

Question

In this problem, $\beta_u$, $w_{u,c}$ (a vector of complex elements), $x_u$ are optimization variables.

Now,

$||2\sqrt{\frac{\pi_u}{2}}\beta_u; h_{u,c}^{\rm H}w_{u,c}-\frac{1}{2\pi_u}x_u-1||_2\le h_{u,c}^{\rm H}w_{u,c}-\frac{1}{2\pi_u}x_u-1$

is a valid convex constraint, hyperbolic constraint. Here $h_{u,c}^{\rm H}w_{u,c}$ came for the original expression, $\sqrt{|h_{u,c}^{\rm H}w_{u,c}|^2}$

Now, in my problem, instead of $\sqrt{|h_{u,c}^{\rm H}w_{u,c}|^2}$, I have $\sqrt{\sum_{c=1}^{C}|h_{u,c}^{\rm H}w_{u,c}|^2}$.

$\textbf{How can I put in the form of hyperbolic constraint as above?}$

$||2\sqrt{\frac{\pi_u}{2}}\beta_u; \sqrt{\sum_{c=1}^{C}|h_{u,c}^{\rm H}w_{u,c}|^2}-\frac{1}{2\pi_u}x_u-1||_2\le \sqrt{\sum_{c=1}^{C}|h_{u,c}^{\rm H}w_{u,c}|^2}-\frac{1}{2\pi_u}x_u-1$

Answer 1

You are asking about representations of $\sqrt{a^2 + b^2} \leq b$.

It is trivially only feasible for $a = 0$ and $b\geq 0$. The constraint $b\geq 0$ term will be problematic in your new model as it represents a nonconvex quadratic set.