Some True or False questions on Nonlinear Optimisation (Exam Preparation)

Question

I am currently preparing for a Nonlinear Optimisation exam and am working through some old question papers and came across these True or False questions:

1. When minimizing a convex function over a convex set, the optimal solution is always on the boundary of the set.
2. For a convex optimisation problem with constraints, if a feasible point satisfies the Karush-Kuhn-Tucker conditions, then it is a global optimum.
3. For a nonlinear optimization problem, if Newton's Method converges, then it converges to a local minimum.

My attempt:

1. (I am unsure about this - I have a feeling it is false, but can someone please explain to me why this is true or false?)

2. True provided that we are working with a minimization problem, since we know that if we minimize a convex function $f$ over a convex set $C$, then the local minimum point $\bar{x}_0$ is also a global minimum. If, however, we are maximizing, we require $f$ to be concave and the set $C$ to be convex. Then the local maximum point $\bar{x}_0$ will also be a global maximum.

3. According to Wikipedia, this is true. I honestly do not know why though. Can anyone please explain to me?

My apologies if this question seems unfit for Math.SE, but I am asking these questions to ensure that I get as good an understanding of the work and the way these things work, instead of simply memorising theorems and proofs without any real understanding.

Answer 1

1. Your feeling is right. Try to figure out a counterexample. Think of something simple like one-dimensional parabolic.
2. True, but explanation is unclear to me. How do you know that the KKT is a local minimum? The sufficiency of KKT for convex problems is typically proved via Lagrangian $L$, which turns out to be convex too, so KKT point is a stationary point for $L$, finalizing with some estimations/inequalities between $L$ and the function.
3. Hmmm... Try Newton on $f(x)=-x^2$ starting at, for example, $x_0=1$.