Non-Convex Constraints for Classification Problems

Question

I am willing to create a hypothetical non-convex constraints for the purpose of practising nonlinear classification using an algorithm. I thought of such constraints in the form: $x^TAx + Bx \leq c$.

I am curious if this qualifies for non-convex constraints, and if the matrices $A$ and $B$ are necessarily PSD. Or could I possibly have more than this constraints?

I would like if someone explains this or refers me to any text/paper I could read up. My Mathematics is kind of rusty at the moment.

Answer 1

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.