The only conditions that I know of under which it's absolutely known that gradient flow will converge to a minimum energy state is when the initial configuration is planar. Zheng-Xu He proved this for his renormalized $1/r^2$ potential 'Mobius' energy in The Euler-Lagrange equation and heat flow for the Möbius energy (CPAM 53, 2000). We proved a similar theorem for 'repulsive' energies on planar polygons in An energy-driven approach to linkage unfolding (SOCG 2004).
For a nonplanar configuration, you're right to think that there's a general theorem that ``simulated annealing has a positive probability of converging to the global minimum" which applies to these problems, but it's ineffective in practice. The configuration space is in practice very high dimensional (3 x number of vertices) and reducing an energy functional such as ropelength generally requires a coordinated global motion of these vertices, so it's quite rare to generate such a move randomly. AFAIK, nobody knows how to estimate the probability of convergence under these circumstances, so there's no hard information on how long to run an annealer.
FWIW, RidgeRunner really isn't an annealer: it generates 'coordinated motions' for tightening deterministically using a linear algebra algorithm to deflect the gradient of length for a polygon into a ropelength-decreasing direction using the active distance and curvature constraints. The algorithm is described in Knot Tightening By Constrained Gradient Descent (Experimental Math, 2011).
You're welcome to play with RidgeRunner yourself if you'd like to try tightening: the software can be downloaded at link text.
