Dual cone and polar cone: Difference between revisions

Revision as of 16:22, 26 March 2012

Latest revision as of 01:29, 13 August 2025

(44 intermediate revisions by 30 users not shown)

Line 1:

{{Short description|Concepts in convex analysis}}

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

[[ImageFile:PolarDual cone illustration1illustration.svg|right|thumb|A set <math>''C</math>'' and its polardual cone <math>''C^o</math>. The dual cone and the polar cone are symmetric to each other with respect to the origin{{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''' <math>C^* </math> of a [[subset]] <math>C</math> in a [[linear space]] <math>X</math>, e.g. [[Euclidean space]] <math>\mathbb R^n</math>, with [[topological]] [[dual space]] <math>X^*</math> is the set

=== In a vector space ===

The '''dual cone''' <math>''C^{{sup|* </math>}}'' of a [[subset]] <math>''C</math>'' in a [[linear space]] <math>''X</math>'' over the [[real numbers|real]]s, e.g. [[Euclidean space]] '''R'''<mathsup>\mathbb R^''n''</mathsup>, with [[topological]] [[dual space]] <math>''X^{{sup|*</math>}}'' 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>.

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

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) ===

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 ===

WhenUsing <math>this latter definition for ''C{{sup|*}}'', </math>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=9780521833783978-0-521-83378-3 | url=httphttps://wwwweb.stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf#page=65 |format=pdf|accessdateaccess-date=October 15, 2011|pages=51-5351–53}}</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.

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

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

#''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.

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

*<math>C_1 \subseteq C_2</math> implies <math>C_2^* \subseteq C_1^*</math>.

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

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

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

== Self-dual cones ==

A cone is said to be ''self-dual'' if <math>C = C^* </math>. The nonnegative [[orthant]] of <math>\mathbb{R}^n</math> and the space of all [[positive semidefinite matrix|positive semidefinite matrices]] are self-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>

==Polar cone==

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.

[[Image: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.]]

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

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

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.

== Polar cone ==

[[ImageFile:Polar cone illustration.svg|right|thumb|The polar of the closed convex cone <math>''C</math>'' is the closed convex cone ''C<mathsup>C^o,</mathsup>'', 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|orig-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 cone is equal to the negative of the dual cone, i.e. ''C<mathsup>C^o=-C^*</mathsup>'' = −''C{{sup|*}}''.

For a closed convex cone <math>''C</math>'' in <math>''X</math>'', the polar cone is equivalent to the [[polar set]] for <math>''C</math>''.<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.

| coauthors = Yang, X.Q.

| title = Duality in optimization and variational inequalities

| publisher = London; New York: Taylor & Francis

| date = 2002

| pages =

| isbn = 0415274796

}}

*{{cite book

| last = Boltyanski

| first = V. G.

| authorlinkauthor-link= Vladimir Boltyansky

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

| title = Excursions into combinatorial geometry

| publisher = New York: Springer

| dateyear = 1997

| pagesisbn = 3-540-61341-2

| isbn = 3540613412

}}

*{{cite book

| last = Goh

| first = C. J.

| coauthors author2= Yang, X.Q.

| title = Duality in optimization and variational inequalities

| publisher = London; New York: Taylor & Francis

| dateyear = 2002

| isbn = 04152747960-415-27479-6

}}

* {{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

| dateyear = 2000

| pagesisbn = 0-8218-1990-9

| isbn = 0821819909

}}

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

{{Ordered topological vector spaces}}

[[Category:MathematicalConvex optimizationanalysis]]

[[Category:Convex geometry]]

[[Category:Linear programming]]

[[Category:Convex analysis]]

[[nl:Duale kegel]]