I know that $||x||^2_2$ is a convex function. I want to check if the following constraint is convex.
$$\sum^N_i \alpha_i -\sum^N_i \frac{||\mathbf{x}_i-\mathbf{z}||_2^2}{\beta} \alpha_i \leq constant $$
Where, $\beta > 0$ is contant, $\alpha_i's$ are constant can be both negative and positive, $\mathbf{x}_i's \in \mathbb{R}^n $ are constant vectors and $\mathbf{z}\in \mathbb{R}^n $ is variable.
Is there any established convex relaxation if it is not convex?
No, it is not. Just take $N=1, \alpha _1=1=\beta$ and $x_1=0$.