Dual cone and polar cone: Difference between revisions

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{{Short description|Concepts 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.]]

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

== Dual cone ==

The '''dual cone''' ''C*'' of a [[subset]] ''C'' in a [[linear space]] ''X'', e.g. [[Euclidean space]] '''R'''^''n'', with [[topological]] [[dual space]] ''X*'' is the set

=== In a vector space ===

The '''dual cone''' ''C{{sup|*}}'' of a [[subset]] ''C'' in a [[linear space]] ''X'' over the [[real numbers|real]]s, e.g. [[Euclidean space]] '''R'''^''n'', with [[topological]] [[dual space]] ''X{{sup|*}}'' is the set

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

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^*''</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>.

=== In a Hilbert space (internal dual cone) ===

''C*'' is always a [[convex cone]], even if ''C'' is neither [[convex set|convex]] nor a [[linear cone|cone]].

Alternatively, many authors define the dual cone in the context of a real [[Hilbert space]], (such as '''R'''^''n'' equipped with the Euclidean inner product) to be what is sometimes called the ''internal dual cone''.

:<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=httphttps://wwwweb.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''.

#''y'' and ''C'' lie on the same side of that supporting hyperplane.

*''C{{sup|*}}'' is [[closed set|closed]] and convex.

*''C''<sub>1</submath>C_1 ⊆\subseteq ''C''<sub>2C_2</submath> implies <math>C_2^* \subseteq C_1^*</math>.

*If ''C'' has nonempty interior, then ''C{{sup|*}}'' is ''pointed'', i.e. ''C*'' contains no line in its entirety.

*If ''C'' is a cone and the closure of ''C'' is pointed, then ''C{{sup|*}}'' has nonempty interior.

*''C{{sup|**}}'' is the closure of the smallest convex cone containing ''C''. (a consequence of the [[hyperplane separation theorem]])

== Self-dual cones ==

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 than 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 with spherical base in '''R'''^''n'' is equal to its internal dual.

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>

The nonnegative [[orthant]] of '''R'''^''n'' and the space of all [[positive semidefinite matrix|positive semidefinite matrices]] are self-dual, as are the cones with ellipsoidal base (often called "spherical cones", "Lorentz cones", or sometimes "ice-cream cones"). So are all cones in '''R'''³ whose base is the convex hull of a regular polygon with an odd number of vertices. A less regular example is the cone in '''R'''³ whose base is the "house": the convex hull of a square and a point outside the square forming an equilateral triangle (of the appropriate height) with one of the sides of the square.

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 from 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.

The nonnegative [[orthant]] of '''R'''^''n'' and the space of all [[positive semidefinite matrix|positive semidefinite matrices]] are self-dual, as are the cones with ellipsoidal base (often called "spherical cones", "Lorentz cones", or sometimes "ice-cream cones"). So are all cones in '''R'''³ whose base is the convex hull of a regular polygon with an odd number of vertices. A less regular example is the cone in '''R'''³ whose base is the "house": the convex hull of a square and a point outside the square forming an equilateral triangle (of the appropriate height) with one of the sides of the square.

So are all cones in '''R'''³ whose base is the convex hull of a regular polygon with an odd number of vertices.

A less regular example is the cone in '''R'''³ whose base is the "house": the convex hull of a square and a point outside the square forming an equilateral triangle (of the appropriate height) with one of the sides of the square.

== Polar cone ==

==Polar cone==

[[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>

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 ==

* [[Bipolar theorem]]

* [[Polar set]]

== References ==

{{Reflist}}

==Bibliography==

*{{cite book

| last = Goh

| first = C. J.

|author2=Yang, X.Q.

| title = Duality in optimization and variational inequalities

| publisher = London; New York: Taylor & Francis

| year = 2002

| pages =

| isbn = 0-415-27479-6

}}

*{{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

}}

*{{cite book

| last = Goh

| first = C. J.

|author2=Yang, X.Q.

| title = Duality in optimization and variational inequalities

| publisher = London; New York: Taylor & Francis

| year = 2002

| isbn = 0-415-27479-6

}}

* {{Narici Beckenstein Topological Vector Spaces|edition=2}} <!-- {{sfn | Narici | 2011 | p=}} -->

*{{cite book

| last = Ramm

| first = A.G.

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

| 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}}

[[Category:Mathematical optimization]]

[[Category:Convex analysis]]

[[Category:Convex geometry]]

[[Category:Linear programming]]

[[Category:Convex analysis]]