Dual cone and polar cone: Difference between revisions

Revision as of 13:08, 4 September 2012

Revision as of 09:08, 4 October 2012

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<math>C^* </math> is always a [[convex cone]], even if <math>C </math> 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 <math>\mathbb{R}^n</math> equipped with the Euclidean inner product) to be what is sometimes called the <i>''internal dual cone</i>''.

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

Using this latter definition for <math>C^*</math>, we have that

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=978-0-521-83378-3|url=http://www.stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf|format=pdf|accessdate=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.

*<math>C^* </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^* </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^* </math> has nonempty interior.

*<math>C^{**} </math> is the closure of the smallest convex cone containing <math>C </math>.

==Self-dual cones==

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<math>\langle . , . \rangle</math> such that the

internal dual cone relative to this inner product

is equal to <math>C</math>.<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 <math>\mathbb{R}^n</math> with ellipsoidal base self-dual, because the inner product can be changed to make the base spherical, and a with spherical base

in <math>\mathbb{R}^n</math> is equal to its internal dual.

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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=978-0-691-01586-6|pages=121-122121–122}}</ref>

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

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| title = Duality in optimization and variational inequalities

| publisher = London; New York: Taylor & Francis

| dateyear = 2002

| pages =

| isbn = 0-415-27479-6

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| title = Excursions into combinatorial geometry

| publisher = New York: Springer

| dateyear = 1997

| pages =

| isbn = 3-540-61341-2

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| title = Operator theory and its applications

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

| dateyear = 2000

| pages =

| isbn = 0-8218-1990-9

}}

[[Category:Mathematical optimization]]