Dual cone and polar cone

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A set C and its dual cone C * .
A set C and its dual cone C * .
A set C and its polar cone Co. The dual cone and the polar cone are symmetric to each other respect to the origin.
A set C and its polar cone Co. The dual cone and the polar cone are symmetric to each other respect to the origin.

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

[edit] Dual cone

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

C^* = \left \{y\in \mathbb R^n: y \cdot x \geq 0 \quad \forall x\in C \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 * * 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{R}^n and the space of all positive semidefinite matrices are self-dual.

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

[edit] Polar cone

The polar of the closed convex cone C is the closed convex cone Co, and vice-versa.
The polar of the closed convex cone C is the closed convex cone Co, and vice-versa.

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

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

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

[edit] References

  • Goh, C. J.; Yang, X.Q. (2002). Duality in optimization and variational inequalities. London; New York: Taylor & Francis. ISBN 0415274796. 
  • Boltyanski, V. G.; Martini, H., Soltan, P. (1997). Excursions into combinatorial geometry. New York: Springer. ISBN 3540613412. 
  • Ramm, A.G.; Shivakumar, P.N.; Strauss, A.V. editors (2000). Operator theory and its applications. Providence, R.I.: American Mathematical Society. ISBN 0821819909.