I have no clue who actually reads this space, so I'll throw this out there. Help would be much appreciated.
Okay, we have Berge's Maximum Theorem, which basically says that when the objective function is well-behaved, we can be assured of structure on the value function (continuity) and on the maximizer correspondence (upper hemi-continuity, compact-valuedness).
If anyone knows of a theorem that works in the complex field as opposed to R^n (not that important) and guarantees both upper and lower hemi-continuity on the maximizer correspondence (rather more important), please take the time to post it.
The problem seems to be that the maximum theorem makes all the assumptions already, there's nothing left to strengthen.
EDIT: Upon scant reflection, there's something I'm missing. I believe the objective is convex. Otherwise strict quasiconcavity is a sufficient condition.