Dual cone and polar cone: Difference between revisions

Revision as of 03:46, 19 October 2019

Revision as of 02:41, 19 January 2020

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:<math>C^*_\text{internal} := \left \{y\in X: \langle y , x \rangle \geq 0 \quad \forall x\in C \right \}.</math>

Using this latter definition for ''C{{sup|*}}'', we have that when ''C'' is a cone, the following properties hold:<ref name="Boyd">{{cite book|title=Convex Optimization | first1=Stephen P. |last1=Boyd |first2=Lieven|last2=Vandenberghe|year=2004|publisher=Cambridge University Press|isbn=978-0-521-83378-3 | url=httphttps://wwwweb.stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf#page=65 |format=pdf|accessdate=October 15, 2011|pages=51–53}}</ref>

* A non-zero vector ''y'' is in ''C{{sup|*}}'' if and only if both of the following conditions hold:

#''y'' is a [[surface normal|normal]] at the origin of a [[hyperplane]] that [[supporting hyperplane|supports]] ''C''.