Consider the problem\begin{equation}\min_{x\in X, y \in Y} f(x,y)\end{equation}Can I solve the problem by iteratively solving the following two sub problems?\begin{equation}x_{k+1} = \arg\min_{x\in X} f(x,y_k)\end{equation}\begin{equation}y_{k+1} = \arg\min_{ y \in Y} f(x_k,y)\end{equation}If yes, is there any books/papers talking about this kind of approach?Thanks
Iteratively minimizing the function
reference request;optimization
It's not hard to show that if $f$ is strongly convex then this approach converges and you can give an upper bound on the number of iterations needed.If $f$ is strictly convex, but not strongly convex, this might converge very slow or even not converge at all.If $f$ is not convex at all, you are likely to end in a local optima rather than the global one.You might also like to take a look at  the Expectationmaximization algorithm which is quite similar, but also deals with unobserved values.Also, there are applications where $f$ is a non-convex function at all with respect to $x,y$, but it is convex w.r.t. each of them (e.g. $f(x,y)=xy$).In these problems, the iterative process you describe is many times used in practice and works quite well.