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Dual cone and polar cone

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Dual cone and polar cone

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

Contents

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 ∗ = { y ∈ X ∗ : ⟨ y , x ⟩ ≥ 0 ∀ x ∈ C } ,

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 internal ∗ := { y ∈ X : ⟨ y , x ⟩ ≥ 0 ∀ x ∈ C } .

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

  • 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 ∗ ⊆ 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. 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

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

    C o = { y ∈ X ∗ : ⟨ y , x ⟩ ≤ 0 ∀ x ∈ C } .

    It can be seen that the polar cone is equal to the negative of the dual cone, i.e. Co = −C*.

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

    References

    Dual cone and polar cone Wikipedia
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