# How can I model this Hyperbolic constraint?

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 $$\sqrt{\sum_{c=1}^{C}|h_{u,c}^{\rm H}w_{u,c}|^2}$$.

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

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

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.