It is well known that a optimization problem can be turned into a decision problem with an extra parameter: e.g. in TSP we are looking for the lowest cost for a tour, the decision version therefore could check whether a a tour exists which costs at most $K$.
Lets assume we are talking about an optimization problem which is either NP or NP-hard. Can the corresponding decision problem ever be in P?
NB: What I am trying to ask is: if a optimization problem cannot be solved in polynomial time, can the corresponding decision problem be solve in polynomial time?
