Iteratively minimizing the function
This is a very popular heuristic method in machine learning, known as alternating minimization. You can easily find tons of papers using it. Often the setting in which it is used is when $f(x,y)$ is convex in $x$ and $y$ separately (so each minimization step can be done via some variant of gradient descent), but not jointly convex. In this case, as RB notes, it is not apriori clear that this method converges to an optimal solution at all. A classical paper by Csiszar and Tusnady gives a general "five point criterion" for convergance to the global optimum, and proves that it holds for $f$ equal to the KL-divergence. More recently, Jain, Nethrapalli and Singhavi, and Hardt have analyzed variants of this approach for the matrix completion problem.