Is the set defined by this inequality convex or not?

1

We have the following inequality

\| \mathbf{x}^T\mathbf{A} + \mathbf{b} \|_2 \leq c| y |

where \mathbf{x} \in \mathbb{C}^n and y\in\mathbb{C} are variables. Matrix \mathbf{A}\in\mathbb{C}^{( n\times (n+1) )} and vector \mathbf{b}\in\mathbb{C}^{( 1\times (n+1) )}.

My questions are:

i) Is the set defined by variables \mathbf{x} and y convex or non-convex?

ii) If it is convex, how can I prove its convexity?

iii) If it is non-convex, how can I deal with its non-convexity? Specifically, is there any way to relax this inequality and make it convex.

2

Please re-read this link which has previously been provided to you.

abs(y) on the RHS of <= makes this non-convex. That is, of course, unless c <= 0, in which case the constraint reduces to x'*A == -b, y == 0.

Whether there is a convex modification of this constraint which makes sense for your intended use is up to you to determine.