Non-Convex Constraints for Classification Problems
Notice that constraint  of the form of $g(x) \le c$ where $g$ is convex is convex.

To see this, consider $g(x_i) \le c$, then for $\lambda \in (0,1)$,

$$g(\lambda x_1 + (1-\lambda)x_2) \le \lambda g(x_1) + (1-\lambda) g(x_2) \le \lambda c + (1-\lambda) c = c$$

To make it non-convex, let $A$ be negative definite or indefinite.