Reference request: optimal solution for linear program is rational

Question

From various lecture notes and such that are floating around, I get the impression that if a linear program has rational coefficients and has a finite optimal solution, then that solution will be rational. This is probably easy to prove, but is there some citeable reference that gives this fact?

Answer 1

This is stated in section 3.7 of "Semidefinite Optimization" by Monique Laurent and Frank Vallentin (Mastermath, Spring 2012). This is technically a lecture note, but takes a formal tone and reads like a book so I would consider it citeable.

Note: it is possible that there are optimal solutions which are not rational, but that there will be at least one optimal solution which is rational.