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# How to find the dual cone of {Ax | x>=0} ?

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#### thisnot

##### Junior Member level 1
dual cone of ax

Can anybody explain how to find the dual cone of {Ax | x >= 0}, where A is an mxn matrix?

##### Full Member level 3
find the dual cone of

Given a cone $K$
$y \in K = \{Ax|x\geq 0\}$
The duality set $K^{\ast}$ is defined as
$x^{\ast} \in K^{\ast}=\{y^{T}x^{\ast}\geq 0$ for all $y \in K \}$
then
$y^{T}x^{\ast}=(Ax)^{T}x^{\ast}=x^{T}(A^{T}x^{\ast})\geq 0$
Since $x \geq 0$, the duality set is
$K^{\ast}=\{A^{T}x^{\ast}\geq0\}$
which is a polyhedral cone (intersections of finite number of halfspaces that have corresponding halfplanes passing through origin.)

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