Finding lexicographic maximum point using a linear program I'm trying to find the lexicographical maximum point of a bounded polyhedron, i.e. I have a set $P = \{x \in \mathbb{R}^n : Ax \leq b\}$ and I'm looking for the lexicographic maximum point of this set.
Note also that every component of $x$ is bounded by a constant. 
I want to solve this problem using a linear program. 
How would you proceed?
 A: There's a proof in this paper by Stef Tijs that the approach Henning suggests of maximizing $x_1 + \epsilon x_2 + \epsilon^2 x_3 + \cdots + \epsilon^{n-1} x_n$, for sufficiently small $\epsilon$, will work.  Unfortunately, the proof is nonconstructive, so it doesn't tell you exactly which value of $\epsilon$ to use.  It just proves that there exists some $\hat{\epsilon}$ such that, for all $\epsilon \leq \hat{\epsilon}$, maximizing $x_1 + \epsilon x_2 + \epsilon^2 x_3 + \cdots + \epsilon^{n-1} x_n$ over $P$ will give you the lexicographic maximum of $P$.  I think Henning's suggestion to let $\epsilon$ be smaller than the ratio between any two nonzero entries of $A$ should work, though.
Yuval's idea of finding the lexicographic maximum by iteratively finding the value of $x_1$, then $x_2$, and so forth should also work.  The disadvantage there is that it might take up to $n$ separate linear programming problems, while Henning's suggestion for small enough $\epsilon$ would find the lexicographic maximum with just one LP.
