Does a constrained nonlinear program only have one KKT point?

Question

I am just wondering whether or not a constrained nonlinear program has only one KKT point?

Intuitively, I think this is wrong and the only thing we can conclude is that any KKT in such program is also a global minimum.

Answer 1

No. The KKT conditions are a local check in the neighborhood of a single point. There may be many other points satisfying the KKT conditions in other disjoint neighborhoods, and therefore one cannot make a claim about global optimality about one such point without potentially checking all others and placing some restrictions on the nonlinear program itself.

Answer 2

Consider the very simple nonlinear program $$ \begin{aligned} \min_{x \in \mathbb{R}}&\quad \sin(\pi x) \\ \text{s.t.} &\quad x^2-3x+2=0 \end{aligned} $$ It has two feasible points $x=1$ and $x=2$, and both are optimal since they yield the same objective value. You are welcome to verify that both are indeed KKT points.