Consider $f:\mathbb{R}^n \rightarrow \mathbb{R}$ convex and $C^1$, $g:\mathbb{R}^m \rightarrow \mathbb{R}^n$ with all $g_i$ convex, $C^1$ and $h:\mathbb{R}^n \rightarrow \mathbb{R}^p$ affine linear and the problem $$ \min_{x \in \mathbb{R}^n} f(x) \\ \mathrm{s.t.} \quad g(x) \le 0, ~h(x) = 0, $$ i.e. a generic convex problem.
It is well known that this type of convexity is not a constrained qualification, i.e. there can be minimal points that are not KKT-points (i.e. $f(x) = x$, $g(x) = x^2$, $h(x) = 0$). So I think it should be possible to construct a problem of this type with exactly 42 KKT-points. Problem is, that all those are global minima and by convexity, there are then infinitely many global minima. So some of them have to be KKT-points and some of them not. Can anyone provide me with an example of such problem or a proof that such problem can't exist?
Thank you in advance.