If there is a "stubborn" stain it may in fact already be a cleaning mark, i.e. damage to the AR coating.
Originally Posted by Dave Dawson
I would agree with the injunction not to clean unnecessarily, even if your cleaning cloth isn't abrasive it might pick up something that is and cause damage.
I find the microfibre cloths sold for optics cleaning are effective and safe on modern camera lens coatings (which are harder and more durable than ones from decades ago).
Blow off any dust with canned air or a blower brush first (read the instructions on canned air carefully).
If you want to use anything liquid you might start with "windex" window cleaner (vers slightly dampen your cleaning cloth only) since this will remove polar, non-polar and mixed soils. You may need to follow up with the highest purity isopropanol you can obtain since windex itself can leave stains.
Wipe from the centre to the edge of the surface.
(BTW I would vote for the no-filter-but-use-a-lens-hood-option)