I have not got any attention/comment on this question yet, even after being in bounty for 4 days. Meanwhile I learned that the problem has trivial solution in higher dimensions (minimum value = 0), but that not seem to be the case in single ($d = 1$) dimension. If this is a myth, I'd like someone to bust it with a counter example or a hint. But I am not getting any attention on that as well. So I wonder if I should consider migrating it to math.SE, if its absolutely sure that it is not suitable here. It is still under bounty.
I am interested in the single dimension case, to obtain a solution if its not trivial.
EDIT : Update
This question was migrated upon my request, and just got an amazing answer over there.