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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.

Dual cone

The dual cone $C^*$ of a subset $C$ in a Euclidean space $mathbb R^n$ is the set

$C^* = left \left\{yin mathbb R^n: y cdot x geq 0 quad forall xin C right \right\},$

where "·" denotes the dot product.

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

When $C$ is a cone, the following properties hold:

• A non-zero vector $y$ is in $C^*$ if and only if $-y$ is the normal of a hyperplane that supports $C$ at the origin.
• $C^*$ 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 the closure of $C$ is pointed, then $C^*$ has nonempty interior.
• $C^\left\{**\right\}$ is the closure of the smallest convex cone containing $C$.

A cone is said to be self-dual if $C = C^*$. The nonnegative orthant of $mathbb\left\{R\right\}^n$ and the space of all positive semidefinite matrices are self-dual.

Dual cones can be more generally defined on real Hilbert spaces.

Polar cone

For a set $C$ in $mathbb R^n,$ the polar cone of $C$ is the set

$C^o = left \left\{yin mathbb R^n: y cdot x leq 0 quad forall xin C right \right\}.$

It is easy to check that $C^o=-C^*$ for any set $C$ in $mathbb R^n,$ and that the polar cone shares many of the properties of the dual cone.

References

• Goh, C. J.; Yang, X.Q. Duality in optimization and variational inequalities. London; New York: Taylor & Francis.
• Boltyanski, V. G.; Martini, H., Soltan, P. Excursions into combinatorial geometry. New York: Springer.
• Ramm, A.G.; Shivakumar, P.N.; Strauss, A.V. editors Operator theory and its applications. Providence, R.I.: American Mathematical Society.

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