cdd uses some heuristics to decide on the order in which the inequalities (or, dually, the points) are processed. There are ways to tweak it, if I remember correctly, but polymake just uses the default behavior. For details on cdd see
http://www.inf.ethz.ch/personal/fukudak/cdd_home/.
If you find it difficult to solve a specific convex hull problem, you can also try the other implementations: beneath_beyond, lrs and ppl (where the latter is only available in recent betas). All of them have pros and cons. beneath_beyond is original polymake code, lrs (like cdd) is shipped with polymake under the GPL. For ppl we only provide bare interfaces, the code itself must be installed extra (but this is available as a package for most distros). ppl implements the same algorithm as cdd and is often faster.
We are in the middle of writing a paper with massive tests on these algorithms resulting in a few "rules of thumb". Stay tuned.
