Dual cone and 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^* of a subset C in a linear space X over the reals, e.g. Euclidean space Rⁿ, with dual space X^* 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^*, i.e. $\langle y,x\rangle =y(x)$ .

C^* 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\}

,^[1]

which is the polar of the set -C.^[1] 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^*, we have that when C is a cone, the following properties hold:^[2]

A non-zero vector y is in C^* 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^* is closed and convex.
$C_{1}\subseteq C_{2}$ implies $C_{2}^{*}\subseteq C_{1}^{*}$ .
If C has nonempty interior, then C^* 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^* has nonempty interior.
C^** 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.^[3] 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

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^[4]

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^*.

For a closed convex cone C in X, the polar cone is equivalent to the polar set for C.^[5]

References

^ ^a ^b Schaefer & Wolff 1999, pp. 215–222.
^ Boyd, Stephen P.; Vandenberghe, Lieven (2004). Convex Optimization (pdf). Cambridge University Press. pp. 51–53. ISBN 978-0-521-83378-3. Retrieved October 15, 2011.
^ Iochum, Bruno, "Cônes autopolaires et algèbres de Jordan", Springer, 1984.
^ Rockafellar, R. Tyrrell (1997) [1970]. Convex Analysis. Princeton, NJ: Princeton University Press. pp. 121–122. ISBN 978-0-691-01586-6.
^ Aliprantis, C.D.; Border, K.C. (2007). Infinite Dimensional Analysis: A Hitchhiker's Guide (3 ed.). Springer. p. 215. doi:10.1007/3-540-29587-9. ISBN 978-3-540-32696-0.

Bibliography

Boltyanski, V. G.; Martini, H.; Soltan, P. (1997). Excursions into combinatorial geometry. New York: Springer. ISBN 3-540-61341-2.
Goh, C. J.; Yang, X.Q. (2002). Duality in optimization and variational inequalities. London; New York: Taylor & Francis. ISBN 0-415-27479-6.
Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
Ramm, A.G. (2000). Shivakumar, P.N.; Strauss, A.V. (eds.). Operator theory and its applications. Providence, R.I.: American Mathematical Society. ISBN 0-8218-1990-9.
Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.

[FOOTNOTESchaeferWolff1999215–222-1] Schaefer & Wolff 1999, pp. 215–222.

[Boyd-2] Boyd, Stephen P.; Vandenberghe, Lieven (2004). Convex Optimization (pdf). Cambridge University Press. pp. 51–53. ISBN 978-0-521-83378-3. Retrieved October 15, 2011.

[3] Iochum, Bruno, "Cônes autopolaires et algèbres de Jordan", Springer, 1984.

[Rockafellar-4] Rockafellar, R. Tyrrell (1997) [1970]. Convex Analysis. Princeton, NJ: Princeton University Press. pp. 121–122. ISBN 978-0-691-01586-6.

[5] Aliprantis, C.D.; Border, K.C. (2007). Infinite Dimensional Analysis: A Hitchhiker's Guide (3 ed.). Springer. p. 215. doi:10.1007/3-540-29587-9. ISBN 978-3-540-32696-0.

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