Consider the unconstrained non-convex optimization problem:
$$\min\limits_{x,y} f(x,y)$$
Suppose that for fixed $x$, the function $y \mapsto f(x,y)$ is convex. In this case, I believe there are two ways to solve this problem:
Just solve the optimization for both variables $x, y$
Obtain the implicit equation $y^* = h(x)$, which is the solution of $$\min\limits_{y} f(x,y)$$ for fixed $x$ — relatively easy and unique from the convex property, and then solve $$\min\limits_{x} f(x,h(x))$$
I'm curious how these two methods compare — which is generally preferable, and in what cases do individual methods have merit? Also, are there any useful references related to this?