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In this question, it is said that a cone $K$ induces (1) a partial ordering on a set $S$; and (2) a set of generalized inequalities.

What does it mean for a cone to induce a partial ordering on a set? Is there any geometric intuition for this?

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jjjjjj
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  • The answerer defines it: $\displaystyle y \preceq_K x \mathop{\iff}^{\small{\text{def}}} x - y \in K$. – Git Gud Jan 04 '17 at 22:39
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    Thanks, can you provide some intuition on why it is defined this way? I'm having difficulty visualizing it / understanding it. – jjjjjj Jan 04 '17 at 22:56

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Update: p.84 in Dattorro's CVX book provides a nice explanation along with a great visual on p.146.

From what I gather, a cone $K$ induces a partial order by specifying which particular points can be compared (ordered). Points comparable to a particular point $x$ are those points $C$ that fall within the cone $K$ shifted to vertex $x$.

jjjjjj
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Cannot comment, but note that you can also define a generalized inequality w.r.t. to a point not in the original cone. For example, all points less than some particular point $y$ belong to the cone $y - K$ if $K$ is the original cone (and we can say $y \succeq x$ for $x \in w-K$).

dunno
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