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I'm reading Papadimitriou et al'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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maybe it's more or less... –  draks ... Dec 15 '13 at 23:26
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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. –  GEdgar Dec 15 '13 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". –  mjqxxxx Dec 16 '13 at 5:33
    
And such variables can be eliminated by letting $x_j=x_j'-x_j''$ with $x_j'\ge 0$ and $x_j''\ge 0$. –  mjqxxxx Dec 16 '13 at 5:36
    
@mjqxxxx That's a great link! Can you post your comment as an answer? –  JasonMond Dec 16 '13 at 13:14
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1 Answer

up vote 2 down vote accepted

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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