I don't know the terminology, so the title can be confusing. Let me explain here.
- We would like to find the optimal solution in $S$. Suppose some external theory suggests that there must be an optimal solution in $T\subset S$. Since there may be multiple optimal solutions, there may also be an optimal solution in $S \setminus T$, but if what we only care is to achieve the maximum, then failing to find all the optimal solutions is not a problem. So we can search only in $T$. Is there a name for this simplification method?
- The optimal solution is contingent. That is, the optimal solution depends on a condition that we don't know ex-ante. But some external theory suggests that whenever $a$ is the optimal solution, $b$ is also the optimal solution. Then an algorithm is that ignoring $a$, after knowing the condition, calculate the payoff of $b$ and other options, then select the optimal one among them.
In sum, in either case, we rule out some options in the feasible set, knowing at least one optimal solution is still in the remaining set and only search within the remaining set.
I am a layman for operations research. So the description may be messy. Let me know if you need more clarification.