Can I get a little more info on the second bullet point about why we care? What does initialization mean here? Thanks.
I think initialization here means where you start the optimization, which is very "important" in the nonconvex domain.
That's right. For a (strongly) convex function, we can start at any point x and eventually reach the global minimum by doing steepest descent. In the case of a nonconvex function, we can get stuck in a local minimum if we choose the wrong initialization.
Are there subclasses of concave optimizations that are easy to solve as well?
@A-star Maximizing a concave objective function (subject to certain constraints) is effectively the same as minimizing the negated version of that same objective function. Any concave optimization function can be turned into a convex optimization function, and so they are equally easy to solve.