The see-saw algorithm is a method used to optimize a function $f(x,y)$. The method works as follows. Suppose we have a function $f$ with multi variables to be optimized, let's say minimize it, and we group those variables into two groups(we consider only two variables $x$ and $y$ here for simplicity). We then optimize over one group of variables while keeping another group fixed. We iteratively execute the above process until meeting some conditions required.
My question is, are there any guarantees about the success of the algorithm so that we can find global optimal such as the function $f$ must be convex? For example, we can consider $f(x,y)=x^2y$ which can be easily optimized when we fix $x$ to optimize over $y$ or to optimize $x$ while fixing $y$. But when we fix y, let's say y=1, to optimize x, we find optimal to be x=0 and then we are stuck at the value 0. In fact, we can reach a negative number. So how to guarantee the performance of the algorithm?
The answer to the problem is provided in this paper.