- The user asks only about variable ordering. Of course there is a symmetry here: the same question can be asked for equations. Well, equations correspond to slack variables, so this is essentially the same.
- A different ordering may (will?) cause a different solution path, so you can expect different solution times and iteration counts. Of course, this is in a very unpredictable way, so not something we can exploit easily. Especially in MIPs we have this concept of performance variability : minor changes in the model can cause significant different solution times.
- Why would you really worry about the ordering? There are three cases I can think of: (1) access variables by index number [I tend to use higher-level tools where this is not needed], (2) when implementing some decomposition scheme [same: in higher-level tools we can index by names], or (3) when plotting the non-zero elements in the A matrix [I never do this; I find this is not adding much insight into the model]. In practice, I never worry about the ordering. We have already many things to worry about when building LP/MIP models; this should not be one of them.
I don't understand why one would want to spend time on this.
- Andrea Lodi, Andrea Tramontani. Performance Variability in Mixed-Integer Programming. In INFORMS TutORials in Operations Research. Published online: 14 Oct 2014; 1-12, https://pubsonline.informs.org/doi/abs/10.1287/educ.2013.0112