Why can we solve eigenvalue problems which are non-convex by Lagrange multiplier methods and get global minima?

Question

while reading the paper "Some Modified Matrix Eigenvalue Problem" by Golub this doubt occurred to me. there he writes that we can minimize $x^TAx$ subject to $x^TBx=1, Cx=0$

As far as I understand Lagrange multiplier give us global minima only when the problem is convex however this problem is non-convex ($x^TBx=1$ is not affine). Could someone clarify ?

Answer 1

If you use Lagrange multiplier technique, then you get a (maybe) long list of candidates for global minima. Then you can evaluate the function to minimize over these set of candidates.

However, this method is not very efficient in general. It may work for special cases, where something about the structure of the problem is known.