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| [[File:Dual cone illustration.svg|right|thumb|A set ''C'' and its dual cone ''C*''.]]
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| [[File:Polar cone illustration1.svg|right|thumb|A set ''C'' and its polar cone ''C<sup>o</sup>''. The dual cone and the polar cone are symmetric to each other with respect to the origin.]]
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| '''Dual cone''' and '''polar cone''' are closely related concepts in [[convex analysis]], a branch of [[mathematics]].
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| ==Dual cone==
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| The '''dual cone''' ''C*'' of a [[subset]] ''C'' in a [[linear space]] ''X'', e.g. [[Euclidean space]] '''R'''<sup>''n''</sup>, with [[topological]] [[dual space]] ''X*'' is the set
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| :<math>C^* = \left \{y\in X^*: \langle y , x \rangle \geq 0 \quad \forall x\in C \right \},</math>
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| where ⟨''y'', ''x''⟩ is the duality pairing between ''X'' and ''X*'', i.e. ⟨''y'', ''x''⟩ = ''y''(''x'').
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| ''C*'' is always a [[convex cone]], even if ''C'' is neither [[convex set|convex]] nor a [[linear cone|cone]].
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| Alternatively, many authors define the dual cone in the context of a real Hilbert space, (such as '''R'''<sup>''n''</sup> equipped with the Euclidean inner product) to be what is sometimes called the ''internal dual cone''.
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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>
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| Using this latter definition for ''C*'', 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=http://www.stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf |format=pdf|accessdate=October 15, 2011|pages=51–53}}</ref>
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| * A non-zero vector ''y'' is in ''C*'' if and only if both of the following conditions hold:
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| #''y'' is a [[surface normal|normal]] at the origin of a [[hyperplane]] that [[supporting hyperplane|supports]] ''C''.
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| #''y'' and ''C'' lie on the same side of that supporting hyperplane.
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| *''C*'' is [[closed set|closed]] and convex.
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| *''C''<sub>1</sub> ⊆ ''C''<sub>2</sub> implies <math>C_2^* \subseteq C_1^*</math>.
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| *If ''C'' has nonempty interior, then ''C*'' is ''pointed'', i.e. ''C*'' contains no line in its entirety.
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| *If ''C'' is a cone and the closure of ''C'' is pointed, then ''C*'' has nonempty interior.
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| *''C**'' is the closure of the smallest convex cone containing ''C''.
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| ==Self-dual cones==
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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 than the above definition, which permits a change of inner product. For instance, the above definition makes a cone in '''R'''<sup>''n''</sup> with ellipsoidal base self-dual, because the inner product can be changed to make the base spherical, and a with spherical base in '''R'''<sup>''n''</sup> is equal to its internal dual.
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| The nonnegative [[orthant]] of '''R'''<sup>''n''</sup> 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'''<sup>3</sup> 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'''<sup>3</sup> 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.
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| ==Polar cone==
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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<sup>o</sup>'', 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]]|title=Convex Analysis | publisher=Princeton University Press |location=Princeton, NJ|year=1997|origyear=1970|isbn=978-0-691-01586-6|pages=121–122}}</ref>
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| :<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<sup>o</sup>'' = −''C*''.
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| For a closed convex cone ''C'' in ''X'', the polar cone is equivalent to the [[polar set]] for ''C''.<ref>{{cite book|last=Aliprantis |first=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>
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| == See also ==
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| * [[Bipolar theorem]]
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| * [[Polar set]]
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| ==References==
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| {{Reflist}}
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| *{{cite book
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| | last = Goh
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| | first = C. J.
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| | coauthors = Yang, X.Q.
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| | title = Duality in optimization and variational inequalities
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| | publisher = London; New York: Taylor & Francis
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| | year = 2002
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| | pages =
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| | isbn = 0-415-27479-6
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| }}
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| *{{cite book
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| | last = Boltyanski
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| | first = V. G.
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| | authorlink= Vladimir Boltyansky
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| | coauthors = Martini, H., Soltan, P.
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| | title = Excursions into combinatorial geometry
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| | publisher = New York: Springer
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| | year = 1997
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| | pages =
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| | isbn = 3-540-61341-2
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| }}
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| *{{cite book
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| | last = Ramm
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| | first = A.G.
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| | coauthors = Shivakumar, P.N.; Strauss, A.V. editors
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| | title = Operator theory and its applications
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| | publisher = Providence, R.I.: American Mathematical Society
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| | year = 2000
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| | pages =
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| | isbn = 0-8218-1990-9
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| }}
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| [[Category:Mathematical optimization]]
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| [[Category:Convex geometry]]
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| [[Category:Linear programming]]
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| [[Category:Convex analysis]]
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