There's been a little debate in the local group on whether these two cards can play off of each other or not. I personally can see it going either way.
Ruthless Tactics: Resolve one of your dice, increasing its value by the resource cost showing on it.
Imperial HQ: Before you resolve a die showing a resource cost, you may exhaust this support to decrease that resource cost by 1.
So you play Ruthless Tactics to resolve one of your dice...so the question is, does this trigger the "before you resolve a die" condition on Imperial HQ?