In general, we can't know we have the best unless we check every possibility.
However, many problems have special properties that allow us to do better.
Recall in DPLL, we can argue that a partial assignment is no good if it unsatisfies a clause. The reasoning here is that no matter how we set the remaining variables, we still can't do any better than having the formula unsatisfied.
This suggests that we find ways of proving optimistic bounds on completions based on partial solutions. (Most important theme today.)
