In my locale we have a well-established invasive species...the mesquite tree. Farmers and ranchers spend millions of dollars and huge amounts of time and other resources to attempt to "control" them. They use aerial spraying, broadcast chemicals, chaining, and even "grubbing hoes" on individual plants in an attempt to keep them out of their grasslands and crop fields. The best they can do is to limit the encroachment into relatively small swaths of land. There is absolutely no hope of eradicating the mesquite, for it has unbelieveable reproductive potential and is hardy and resistant toward almost all control methods. Are "control" methods having an effect? Sure...on a very limited basis and only for areas where intense efforts are made consistently. If those efforts should falter for even a brief time, we are back to square one.
The same applies to the lionfish. It's absolutely, positively, and completely impossible to eradicate them from the areas they have moved into. The best that can be hoped for is that numbers can be limited in certain areas utilizing intense effort and manpower. A new avenue of study is what conditions enable the lionfish to encroach and dominate certain environments. IF the answer is that other competing species have been "cleared" by factors such as overfishing, then there is hope. By addressing the impact of unlimited take with no protections for those species, then perhaps those competing species can be restored and the effects of the lionfish invasion might be minimized.
It's a hope, anyway.