[aifer4]

You mentioned that such problems may be solved by solving a linear approximation of the non-linear problem (and I am in no way an expert in non-linear optimization). To the extent that the bottleneck in that approach is solving the resulting linear system, this method offers a speedup. We are also thinking about using similar thermodynamic methods to solve non-linear systems directly, but some of the nice properties of the harmonic oscillator are not present in that case, so it's currently not clear how much (if any) speedup is there.