I finally looked at one of the major source papers, An Update on Monitoring Stellar Orbits in the Galactic Center, Astrophysical Journal.

"We use a fitting code we developed in Mathematica (Wolfram Research, Inc. 2016), which calculates the positions and velocities by explicitly integrating the orbits, and then uses the built-in minimization routines to find the best fitting parameters. These routines are based on standard techniques such as quasi-Newton methods, steepest gradient search, or the Levenberg–Marquardt algorithm. Usually, we apply different search algorithms iteratively, until none is able to improve the minimum further."