I have the following optimization problem:
$$ \min_\mathbf{x}\|\mathbf{a+Bx}\|^2 ~~\text{s.t}~~ \|\mathbf{y+Ax}\|_\infty \leq \beta\|\mathbf{y}\|_\infty ~~,~~ \|\mathbf{x}\|^2 \leq \alpha^2$$
where $\mathbf{a} \in \mathbb{C^{M\times 1}}$, $\mathbf{B} \in \mathbb{C^{M\times N}}, \mathbf{A} \in \mathbb{C^{K\times N}}$ and $\mathbf{y \in \mathbb{C^{K\times 1}}}$. $\alpha, \beta$ are constants.
I was wondering under what class of optimization problems does this problem falls and whether the infinity norm constraint is a convex set. Without the infinity norm constraint, the problem is a least squares with a quadratic constraint problem. Any help is appreciated.