NP-complete problem whose corresponding optimization problem is not NP-hard For this question I will refer to$\ NP-hard$ problems as those that are at least as hard as$ \ NP-complete$ problems. That is, a problem$ \ H$ is$\ NP-hard$ if there is an$ \ NP-complete$ problem$\ G$, such that$\ G$ is reducible to$\ H$ in polynomial time.$\ NP-hard$ problems are not restricted to decision problems and are not necessarily in$\ NP$.

Considering the above, is there any optimization problem$\ L$  such that$ \ L \notin NP-hard $ and whose corresponding decision problem is$ \ NP-complete$?

For example, consider the case for the travelling salesman problem. (TSP)


  Optimization problem: Given a list of cities and the distances between each pair, what is the shortest path that visits each city and returns to the original city? 
  
  Decision problem: Given a list of cities, the distances between each pair and a length$ \ L$, does there exist a path that visits each city and returns to the original city of length at most$\ L$.


It is well known that the decision problem of TSP is$ \ NP-complete$ and its corresponding optimization problem is$\ NP-hard$. 

To sum up, what is an example of a$\ NP-complete$ problem whose corresponding optimization problem lies outside the class$\ NP-hard$? Perhaps, it is$\ EXPTIME$.
Take a look at Bellare and Goldwasser, "The complexity of decision vs. search" SIAM J. of Computing 23:1 (Feb 1994), pp. 97-119. It discusses the issue at length. Perhaps a bit more readable is the class note by Bellare, "Decision versus search".