"...the best option in your case is to rip off the band aid and move all rents to market asap"
@Thomas S. I can't say that I necessarily agree with this. I agree that it is important to not lose money, on the other hand, and I face this issue constantly, there is a personal aspect to real estate that you don't find on a P&L statement. There is no easy answer, but the way we have faced this in the past is...
a) Buy properties that don't require a 65% increase in rent in order to work
b) Raise rents somewhere in the middle where it makes sense for both sides, and continue to increase rents over the years
c) When units eventually go vacant, then it is your opportunity to rent it out at full market rent. If they've lived there for 15 years, maybe this won't happen for a while.
Many people would definitely side with Thomas, but you'll be the one that has to live with kicking someone out of the place they've lived for 15 years so you can make an extra buck.
Again, not an easy problem to face.