Let $A, B$ be two convex and closed subsets of $\mathbb{R}^n$. We would like to the minimum distance between these two sets. i.e., we want to find a solution for the following problem.
$$ \min \{||x-y||~ ~\hbox{for}~ x\in A, y\in B\}$$. 


I would like to see if there is a method to find a solution for this problem in general in the theory of optimization. 



Distance between two sets
You are trying to solve what is known as a *best approximation problem*. 


1. von Neumann's alternating projections **does not** work here (as might have been perhaps suggested above)
2. You can use [Dykstra's projection algorithm](http://en.wikipedia.org/wiki/Dykstra%27s_projection_algorithm), which will find the desired projection (some reformulation will be needed though before you can apply it)
3. See the paper by [Bauschke, Combettes, and Luke](http://people.ok.ubc.ca/bauschke/Research/39.pdf) on *reflection* methods---the references in that paper also put this problem in wider context. The "averaged reflections" method (which is essentially the Douglas-Rachford splitting scheme) often works better / faster than Dykstra's projection method.


