Dual cone and polar cone: Difference between revisions

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{{Short description|conceptsConcepts in convex analysis}}

[[File:Dual cone illustration.svg|right|thumb|A set ''C'' and its dual cone ''C{{sup|*}}''.]]

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

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where <math>\langle y, x \rangle</math> is the [[dual system|duality pairing]] between ''X'' and ''X{{sup|*}}'', i.e. <math>\langle y, x\rangle = y(x)</math>.

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

=== In a topological vector space ===

If ''X'' is a [[topological vector space]] over the real or complex numbers, then the '''dual cone''' of a subset ''C'' ⊆ ''X'' is the following set of continuous linear functionals on ''X'':

:<math>C^{\prime} := \left\{ f \in X^{\prime} : \operatorname{Re} \left( f (x) \right) \geq 0 \text{ for all } x \in C \right\}</math>,{{sfn | Schaefer |Wolff| 1999 | pp=215–222}}

which is the [[polar set|polar]] of the set -''C''.{{sfn | Schaefer |Wolff| 1999 | pp=215–222}}

No matter what ''C'' is, <math>C^{\prime}</math> will be a convex cone.

If ''C'' ⊆ {0} then <math>C^{\prime} = X^{\prime}</math>.

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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>

=== Properties ===

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=https://web.stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf#page=65 |format=pdf|accessdateaccess-date=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''.

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A cone ''C'' in a vector space ''X'' is said to be ''self-dual'' if ''X'' can be equipped with an [[inner product]] ⟨⋅,⋅⟩ such that the internal dual cone relative to this inner product is equal to ''C''.<ref>Iochum, Bruno, "Cônes autopolaires et algèbres de Jordan", Springer, 1984.</ref>

Those authors who define the dual cone as the internal dual cone in a real Hilbert space usually say that a cone is self-dual if it is equal to its internal dual.

This is slightly different thanfrom the above definition, which permits a change of inner product.

For instance, the above definition makes a cone in '''R'''^''n'' with ellipsoidal base self-dual, because the inner product can be changed to make the base spherical, and a cone with spherical base in '''R'''^''n'' is equal to its internal dual.

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[[File:Polar cone illustration.svg|right|thumb|The polar of the closed convex cone ''C'' is the closed convex cone ''C^o'', and vice versa.]]

For a set ''C'' in ''X'', the '''polar cone''' of ''C'' is the set<ref name="Rockafellar">{{cite book|author=[[Rockafellar, R. Tyrrell]]|author-link=Rockafellar, R. Tyrrell|title=Convex Analysis | publisher=Princeton University Press |___location=Princeton, NJ|year=1997|origyearorig-year=1970|isbn=978-0-691-01586-6|pages=121–122}}</ref>

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

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It can be seen that the polar cone is equal to the negative of the dual cone, i.e. ''C^o'' = −''C{{sup|*}}''.

For a closed convex cone ''C'' in ''X'', the polar cone is equivalent to the [[polar set]] for ''C''.<ref>{{cite book|lastlast1=Aliprantis |firstfirst1=C.D.|last2=Border |first2=K.C. |title=Infinite Dimensional Analysis: A Hitchhiker's Guide|edition=3|publisher=Springer|year=2007|isbn=978-3-540-32696-0|doi=10.1007/3-540-29587-9|page=215}}</ref>

== See also ==

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{{Reflist}}

==Bibliography==

*{{cite book

| last = Boltyanski

| first = V. G.

| authorlinkauthor-link= Vladimir Boltyansky

|author2=Martini, H. |author3=Soltan, P.

| title = Excursions into combinatorial geometry

| publisher = New York: Springer

| year = 1997

| pages =

| isbn = 3-540-61341-2

}}

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| publisher = London; New York: Taylor & Francis

| year = 2002

| pages =

| isbn = 0-415-27479-6

}}

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| last = Ramm

| first = A.G.

|editor=Shivakumar, P.N. |editor2=Strauss, A.V.

| title = Operator theory and its applications

| publisher = Providence, R.I.: American Mathematical Society

| year = 2000

| pages =

| isbn = 0-8218-1990-9

}}

* {{Schaefer Wolff Topological Vector Spaces|edition=2}} <!-- {{sfn | Schaefer |Wolff| 1999 | p=}} -->

{{Ordered topological vector spaces}}

{{OrderedTopologicalVectorSpaces}}

[[Category:Convex analysis]]

[[Category:Convex geometry]]

[[Category:Linear programming]]

[[Category:Convex analysis]]