# What is $\gtrless$?

I'm reading Papadimitriou & Steiglitz's Combinatorial Optimization and came across notation I'd never seen before and don't know what it means. The $\LaTeX$ markup for it is \gtrless ($\gtrless$), which took me quite a while to find.

It arises in the formulation of general linear programs in terms of the constraints on the variables:

$$x_j \geq 0 \;\; j \in N\\ x_j \gtrless 0 \;\; j \in \bar{N}$$

It's not "not equals" because there's places in the text where the authors say $x$ can be zero.

• maybe it's more or less... Commented Dec 15, 2013 at 23:26
• Maybe there are two cases ... and also two choices somewhere else as well. Top choice goes with top choice, bottom choice goes with bottom choice. Commented Dec 15, 2013 at 23:35
• According to this blog entry (jingjinyu.wordpress.com/2011/02/06/…), which quotes the same section of the same text, it just means "can be any real number". Commented Dec 16, 2013 at 5:33
• And such variables can be eliminated by letting $x_j=x_j'-x_j''$ with $x_j'\ge 0$ and $x_j''\ge 0$. Commented Dec 16, 2013 at 5:36
• @mjqxxxx That's a great link! Can you post your comment as an answer? Commented Dec 16, 2013 at 13:14

According to this blog entry (jingjinyu.wordpress.com/2011/02/06/…), which quotes the same section of the same text, saying that $x_j \gtrless 0$ just means that $x_j$ can be any real number. And as pointed out in the text, such variables can be eliminated by introducing two non-negative auxiliary variables: $x_j=x_j'-x_j''$ with $x_j'\ge 0$ and $x_j''\ge 0$.