The issue with the lion fish in the Carribbean is they have no natural predators, and they completely devastate juvenile fish populations in the areas they inhabit. They eat juvenile predatory fish as well, so they reduce those in addition to the regular populations.
The documentary does an in depth analysis of the way they take their prey, and they have one of the fastest jaw movements in nature
They can be controlled somewhat by regular harvesting and killing, but they never will be eliminated because they have been found in the caribbean at depths greater than 300 feet, which puts them out of reach of any sustained elimination efforts.
Another issue is that none of the native Carribbean fish that would/could eat them recognizes them as prey, like sharks and groupers. Groupers have eaten sick or injured lionfish, but not healthy ones. One diver was trying to teach a native shark to recognize the lionfish as a prey item, however.
Those are some of the highlights from the Documentary I remember.