This performs much better than the trust region Newton's method for the
actual `circles-in-triangle` problem. (The trust region method performs
better for the simplified problem produced by the conversion bug.)
In previous commits, the `circles-in-triangle` example converged much
more slowly in BigFloat precision than in Float64 precision. This
turned out to be a sign of a bug in the Float64 computation: converting
the Gram matrix using `Float64.()` dropped the explicit zeros, removing
many constraints and making the problem much easier to solve. This
commit corrects the Gram matrix conversion. The Float64 search now
solves the same problem as the BigFloat search, with comparable
performance.
Our formula for the improvement theshold works when the step size is
an absolute distance. However, in commit `4d5ea06`, the step size was
measured relative to the current gradient instead. This commit scales
the base step to unit length, so now the step size really is an absolute
distance.
