Questions tagged [dual-cone]

Use this tag for questions involving dual cones. In convex analysis, the "dual cone" to a set is the collection of all elements that form a "positive angle" with every element in the set. That is, given a set $S$ in a vector space $V$, we define the dual cone by $S^* = {y:\langle x,y \rangle \geq 0 \text{ for all } x \in S}$ (the precise meaning of $\langle \cdot,\cdot\rangle$ depends on the context).

In convex analysis, the "dual cone" to a set is the collection of all elements that form a "positive angle" with every element in the set. That is, given a set $S$ in a vector space $V$, we define the dual cone by $S^* = \{y:\langle x,y \rangle \geq 0 \text{ for all } x \in S\}$ (the precise meaning of $\langle \cdot,\cdot\rangle$ depends on the context).

For more information, see this Wikipedia page.

105 questions
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dual cone is closed

I have been trying to understand why dual of a cone is closed, no matter the cone is closed or not. I know the proof is 'It is because dual cone is an intersection of closed halfspaces.'. I just do not understand how it is linked to the definition…
Haesol
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what is the dual of line (x=-y)

I encountered this question: Find the dual cone of $K = \{(x,y)|(x+y=0)\}$. To find the dual ($K^* = \{y \mid x'y \geq 0 \text{ for all }x \in K \}$), I did the following: \begin{align} (x,-x)(y_1,y_2)'& \geq 0 \\ …