I have tried to put my question in following diagram
I have two domains A and B. I can formulate two optimizations problems 1 and 2. In domain A, efficient algorithms exist for solving these two optimization problems. Moreover they both share a mathematical framework "tool", which is well defined in Domain A.
In domain B, It has been proved that optimization problem 1 is NP-hard. Moreover, I also know that "tool" does not exists in domain B. What can I say about optimization problem 2 in domain B. Can we say that it is NP-HARD as well?
I am not a mathematician, so I am not very familiar with the mathematical rigor that we need here. I will appreciate any help in this regard. Also, can someone can suggest a better title for this question?
Technically, you can't say anything about what would happen in domain B. Depending on the domains, and the optimization problems, the optimization problem 2 could be defined in such a way that it is easy in domain A but NP-hard in domain B, or easy in both domains even though your "tool" as you say doesn't work in domain B.
As far as determining what's going on, you can either try to come up with a new polynomial time algorithm for optimization problem 2 in domain B, or, if your optimization problems 1 and 2 are related, then within domain B, you can try to reduce solving an arbitrary optimization problem 1 instance in domain B into an optimization problem 2 instance in domain B (polynomial time reduction), and this would show problem 2 is NP-hard in domain B.