Can we assume without loss of generality that $m \leq n$ in a linear programming theorem?

Question

If I'm making a specific theorem about linear programming, is it acceptable to assume, without loss of generality, that either $m \geq n$, or, $n \geq m$?

Specifically, I saw this in the book Combinatorial Optimization: Algorithms and Complexity by Christos H. Papadimitriou and Kenneth Steiglitz. In their presentation of the Ellipsoid algorithm, they assume $m \geq n$, and prove the correctness and runtime of the Ellipsoid algorithm under this assumption.

Is this valid? Intuitively, I feel it is, since if $m<n$, we can presumably repeat constraints or add additional "silly" constraints to get $m \geq n$.

I'll also add, would the opposite inequality $m \leq n$ be reasonable to assume?

Answer 1

Yes, either assumption is without loss of generality. If it is violated, you can instead solve the dual linear programming problem, which interchanges the roles of variable and constraint and hence reverses the sense of the inequality that relates $n$ and $m$.