I once read a research paper that stated that their customized nonlinear solver reduced computational time for a particular problem class from days to seconds, i.e., something like an 86400X speedup.
Digging a little deeper, it seems the authors did not notice prior work that solved similar sized instances of a more difficult discrete nonlinear case, using an analogous CPLEX-based approach, in a few seconds to a few hours in the worst case. Even a conservative 'from several minutes to a few seconds' mean-improvement is impressive (~100X faster). After deleting complicating side-constraints and relaxing integrality restrictions, the resulting continuous relaxation can indeed be solved really quickly compared to the original problem.
Amartya Sen recently confessed to pulling numbers out of thin air to grab people's attention, lending credence to the claims of his detractors. I hope O(claims) does not turn into a total marketing game in the future.