I have a question about a system in mixed integer programming I am trying to solve using GLPK though the question mathematical and is software agnostic.
I have converted the whole problem to linear equations and inequalities except for one part. Suppose I have a real number $x$ and a binary variable $b$ (either 0 or 1). How can I "tie them together" such that $x > 0$ if and only if $b = 1$?
This question seems to address a similar question but I am suspicious of the answers since I know ILP in NP-complete and the commentors don't seem to acknowledge it.
$$0.001 b \le x \le U b$$ where I assume $0 \le x \le U$.
I don't see how this particular formulation is related to NP-completeness. If you are worried about the complexity of this don't use a MIP model at all but rather use a heuristic (even this may be overstating things as state-of-the-art MIP solvers actually use lots of heuristics internally).