It is known that for every positive integer $n$ there exists one or more optimal addition chains of minimum length. It is rumored that finding the length of the optimal chain is NP-hard, and the related Wikipedia article only provides methods to calculate relatively short chains but not the optimal chain.
1- What methods exist that can find the optimal addition chain for a given $n$?
2- How fast are these methods and how well do they scale with $n$?
3- Do methods exist that will only calculate the length of the optimal chain and could they be faster than ordinary methods?
4- Could a function that relates to addition chains be recursive?