Can I reopen a discussion and restart the exchange?

It is not possible for an agent to reopen the discussion with a client. In fact, once the exchange with your contact is over, the discussion is in the "Closed" section (to find out how to close a discussion, click here). 

Once the discussion is in the "Closed" section, you cannot reopen it. 

The person you are talking to can, however, start the discussion again. In this case, the exchange that you had previously had with him/her returns to the "Pending" section. Then you just have to accept it and it will go to the "In progress" section. The discussion can then restart.