This problem is more tractable than you might think! Some progress on it was printed in the 2016 and 2017 Sigbovik proceedings:

http://sigbovik.org/2016/proceedings.pdf http://sigbovik.org/2017/proceedings.pdf

If this kind of problem appeals to you, please consider participating in http://golf.horse/, a tangentially related challenge that is probably going to tie into Sigbovik 2019.