Dual cone and polar cone: Difference between revisions

Revision as of 16:55, 3 April 2012

Revision as of 16:13, 11 May 2012

Line 1:

[[ImageFile:Dual cone illustration.svg|right|thumb|A set <math>C</math> and its dual cone <math>C^*</math>.]]

[[ImageFile:Polar cone illustration1.svg|right|thumb|A set <math>C</math> and its polar cone <math>C^o</math>. The dual cone and the polar cone are symmetric to each other with respect to the origin.]]

'''Dual cone''' and '''polar cone''' are closely related concepts in [[convex analysis]], a branch of [[mathematics]].

Line 11:

<math>C^* </math> is always a [[convex cone]], even if <math>C </math> is neither [[convex set|convex]] nor a [[linear cone|cone]].

When <math>C </math> 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=9780521833783978-0-521-83378-3|url=http://www.stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf|format=pdf|accessdate=October 15, 2011|pages=51-53}}</ref>

* A non-zero vector <math>y</math> is in <math>C^*</math> if and only if both of the following conditions hold: (i) <math> y </math> is a [[surface normal|normal]] at the origin of a [[hyperplane]] that [[supporting hyperplane|supports]] <math>C </math>. (ii) <math> y </math> and <math>C </math> lie on the same side of that supporting hyperplane.

*<math>C^* </math> is [[closed set|closed]] and convex.

Line 22:

==Polar cone==

[[ImageFile:Polar cone illustration.svg|right|thumb|The polar of the closed convex cone <math>C</math> is the closed convex cone <math>C^o,</math> and vice-versa.]]

For a set <math>C</math> in <math>X</math>, the '''polar cone''' of <math>C</math> is the set

:<math>C^o = \left \{y\in X^*: \langle y , x \rangle \leq 0 \quad \forall x\in C \right \}.</math><ref name="Rockafellar">{{cite book|author=[[Rockafellar, R. Tyrrell]]|title=Convex Analysis|publisher=Princeton University Press|___location=Princeton, NJ|year=1997|origyear=1970|isbn=9780691015866978-0-691-01586-6|pages=121-122}}</ref>

It can be seen that the polar cone cone is equal to the negative of the dual cone, i.e. <math>C^o=-C^*</math>.

Line 46:

| date = 2002

| pages =

| isbn = 04152747960-415-27479-6

}}

Line 58:

| date = 1997

| pages =

| isbn = 35406134123-540-61341-2

}}

Line 69:

| date = 2000

| pages =

| isbn = 08218199090-8218-1990-9

}}