dual cone and polar cone#Polar cone

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

Dual cone

= In a vector space =

The dual cone C{{sup|*}} of a subset C in a linear space X over the reals, e.g. Euclidean space Rⁿ, with dual space X{{sup|*}} is the set

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

where $\langle y, x \rangle$ is the duality pairing between X and X{{sup|*}}, i.e. $\langle y, x\rangle = y(x)$ .

C{{sup|*}} is always a convex cone, even if C is neither convex nor a 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:

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

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

No matter what C is, $C^{\prime}$ will be a convex cone.

If C ⊆ {0} then $C^{\prime} = X^{\prime}$ .

= In a Hilbert space (internal dual cone) =

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

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

= Properties =

Using this latter definition for C{{sup|*}}, we have that when C is a cone, the following properties hold:{{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|access-date=October 15, 2011|pages=51–53}}

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

y is a normal at the origin of a hyperplane that supports C.
y and C lie on the same side of that supporting hyperplane.

C{{sup|*}} is closed and convex.
$C_1 \subseteq C_2$ implies $C_2^* \subseteq C_1^*$ .
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.Iochum, Bruno, "Cônes autopolaires et algèbres de Jordan", Springer, 1984.

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 Rⁿ with ellipsoidal base self-dual, because the inner product can be changed to make the base spherical, and a cone with spherical base in Rⁿ is equal to its internal dual.

The nonnegative orthant of Rⁿ and the space of all 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 R³ whose base is the convex hull of a regular polygon with an odd number of vertices.

A less regular example is the cone in R³ 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

File:Polar cone illustration.svg

For a set C in X, the polar cone of C is the set{{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}}

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

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.{{cite book|last1=Aliprantis |first1=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}}

References

Bibliography

{{cite book

| last = Boltyanski

| first = V. G.

| author-link= Vladimir Boltyansky

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

| title = Excursions into combinatorial geometry

| publisher = New York: Springer

| year = 1997

| 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}}
{{cite book

| 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

| isbn = 0-8218-1990-9

}}

{{Schaefer Wolff Topological Vector Spaces|edition=2}}

Category:Convex analysis

Category:Convex geometry

Category:Linear programming