Dual cone and polar cone

A set C and its dual cone C*.

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 C* of a subset C in a linear space X, e.g. Euclidean space Rⁿ, with topological dual space X* is the set

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

where <y, x> is the duality pairing between X and X*, i.e. <y, x> = y(x).

C* is always a convex cone, even if C is neither convex nor a 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.

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

Using this latter definition for C*, we have that when C is a cone, the following properties hold:^[1]

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

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

• C* is closed and convex.
• C₁ ⊆ C₂ 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.^[2] 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 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^[3]

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

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.^[4]

See also

• Bipolar theorem
• Polar set

References

1. 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.
2. Iochum, Bruno, "Cônes autopolaires et algèbres de Jordan", Springer, 1984.
3. Rockafellar, R. Tyrrell (1997) [1970]. Convex Analysis. Princeton, NJ: Princeton University Press. pp. 121–122. ISBN 978-0-691-01586-6.
4. 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.

• Goh, C. J.; Yang, X.Q. (2002). Duality in optimization and variational inequalities. London; New York: Taylor & Francis. ISBN 0-415-27479-6.

• Boltyanski, V. G.; Martini, H.; Soltan, P. (1997). Excursions into combinatorial geometry. New York: Springer. ISBN 3-540-61341-2.

• 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.