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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.

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  • $\begingroup$ maybe it's more or less... $\endgroup$
    – draks ...
    Commented Dec 15, 2013 at 23:26
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    $\begingroup$ 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. $\endgroup$
    – GEdgar
    Commented Dec 15, 2013 at 23:35
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    $\begingroup$ 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". $\endgroup$
    – mjqxxxx
    Commented Dec 16, 2013 at 5:33
  • $\begingroup$ And such variables can be eliminated by letting $x_j=x_j'-x_j''$ with $x_j'\ge 0$ and $x_j''\ge 0$. $\endgroup$
    – mjqxxxx
    Commented Dec 16, 2013 at 5:36
  • $\begingroup$ @mjqxxxx That's a great link! Can you post your comment as an answer? $\endgroup$
    – JasonMond
    Commented Dec 16, 2013 at 13:14

1 Answer 1

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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 $x_j''\ge 0$.

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