# Dual cone

The **dual cone** is a special cone that can be assigned to any cone. For example, it plays a role in the duality statements of the Lagrange duality in mathematical optimization . It is closely related to the **polar cone** .

## definition

### In Hilbert rooms

A Hilbert space (i.e. a complete vector space with a scalar product ) and a cone in this vector space are given. Then is called the set assigned to the cone

the dual cone of . These are then all vectors which enclose an angle of *at most* 90 ° with *all* elements of the cone . The dual cone is sometimes also referred to as or .

### General case

If the dual space of and is a cone in , then the *dual cone is* defined by

It denotes the dual pairing , that is, it applies .

### comment

Sometimes the first form of definition is already used in incomplete prehilbert spaces in order to be able to understand the resulting sets as a cone in the original space .

## Related terminology

### Polar cone

The concept of the *polar cone can be* formulated analogously :

In a Hilbert space the following applies:

That is the set of all vectors that have an angle of *at least* 90 ° with all cone elements and therefore applies

For both versions of the definition, the relationship results in the respective vector space. This can also be used as a definition.

### Self dual cone

A cone is called self-dual if true.

### comment

Occasionally the dual cone is defined like the polar cone and vice versa, here the literature is ambiguous. It is therefore important to note the direction of the inequality.

## Examples

Considering in provided with the standard scalar the cone with , the dual cone is the right half-plane . For is , so is and this should be for everyone , therefore it must be.

According to the above identity, the polar cone is then the left half-plane.

If one provides the with the scalar product , where the symmetric positive definite matrix

is, so is the dual cone

- .

This is the half-plane that is bounded by the straight line and contains the first quadrant. The scalar product used is therefore decisive for generating the dual (and polar) cone.

An example of a self-dual cone is .

## properties

- The dual and polar cones are convex , regardless of whether or not the original cone already had this property.
- If a topological vector space - with the topological dual space - then the polar and dual cone are always closed .

## literature

- Boyd, Stephen; Vandenberghe, Lieven (2004). Convex Optimization. Cambridge University Press. ISBN 978-0-521-83378-3 . ( online )
- Florian Jarre, Josef Stoer:
*Optimization.*Springer, Berlin 2004, ISBN 3-540-43575-1 . - Peter Knabner , Wolf Barth : Lineare Algebra. Basics and applications (= Springer textbook ). Springer Spectrum, Berlin et al. 2013, ISBN 978-3-642-32185-6 .