"Unusual" coupling between a decision problem and a corresponding optimization problem There seems to usually be a tight connection between decision problems and (corresponding) optimization problems in general. However, is this always the case? 


  Are there examples where the typical "tight coupling" between a decision problem and the correponding optimization problem breaks down or behaves in an unusual way, e.g. have significantly different complexity?


Or, maybe there is a case where there is a cluster of problems that are all closely related, but the "best" or "definitive" version is not obvious or apparent? Also, I am looking for any survey or broad overview or discussion of this apparent basic connection between decision and optimization problems.

A similar question was asked here, but the answers were highly theoretical and it did not seem to yield any specific or tangible examples.
The answer does indeed depend on how you define a "corresponding" problem.

For example, it is well known 2-SAT is easy ("Given a 2-SAT formula, is it satisfiable?"). However, the optimization problem MAX-2-SAT ("Given a 2-SAT formula, find an assignment maximizing the number of clauses") is NP-hard. Even MIN-2-SAT is NP-hard.

For a given decision problem, you can have "corresponding" optimization problems which are either easy or hard, depending on whether we consider a minimization or a maximization problem. The st-connectivity problem is mentioned in the comments: deciding the existence of such a path is easy, finding a shortest st-path is easy, but finding a longest st-path is hard. Another graph theoretic example is with cuts: computing a minimum cut is easy, but computing a maximum cut is again hard.

You also have decision problems that don't have a corresponding optimization problem, such as primality testing.