It is well-know that $\mathcal{P}$ and $\mathcal{NP}$ are classes of decision problems. I have proved that the decision version of my problem is $\mathcal{NP}$-complete. How to show that the optimization version of my problem is also $\mathcal{NP}$-hard?
I in advance thank you for any suggestions!
A polynomial time algorithm to solve the optimization version of the problem would also solve the decision version in polynomial time (just compare the optimal value to the target value for the decision problem.) Thus if the decision problem is NP-Complete, than the optimization problem is NP-Hard.