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 ${\displaystyle V}$ ${\displaystyle \langle .;.\rangle }$ ${\displaystyle {\mathcal {K}}}$

${\displaystyle \operatorname {dual} ({\mathcal {K}})=\{y\in V|\forall x\in {\mathcal {K}}\colon \langle y;x\rangle \geq 0\}}$

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 . ${\displaystyle {\mathcal {K}}}$ ${\displaystyle {\mathcal {K}}^{*}}$ ${\displaystyle {\mathcal {K}}^{D}}$

General case

If the dual space of and is a cone in , then the dual cone is defined by ${\displaystyle V^{*}}$ ${\displaystyle V}$ ${\displaystyle {\mathcal {K}}}$ ${\displaystyle V}$

${\displaystyle \operatorname {dual} ({\mathcal {K}}):=\{y^{*}\in V^{*}|\forall x\in {\mathcal {K}}\colon \langle y^{*};x\rangle \geq 0\}}$

It denotes the dual pairing , that is, it applies . ${\displaystyle \langle .;.\rangle }$ ${\displaystyle \langle y^{*};x\rangle :=y^{*}(x)}$

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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 . ${\displaystyle V}$

Related terminology

Polar cone

The concept of the polar cone can be formulated analogously :

${\displaystyle \operatorname {pol} ({\mathcal {K}}):=\{y^{*}\in V^{*}|\forall x\in {\mathcal {K}}\colon \langle y^{*};x\rangle \leq 0\}}$

In a Hilbert space the following applies:

${\displaystyle \operatorname {pol} ({\mathcal {K}})=\{y\in V|\forall x\in {\mathcal {K}}\colon \langle y;x\rangle \leq 0\}}$

That is the set of all vectors that have an angle of at least 90 ° with all cone elements and therefore applies ${\displaystyle {\mathcal {K}}\cap \operatorname {pol} ({\mathcal {K}})=\{0_{V}\}}$

For both versions of the definition, the relationship results in the respective vector space. This can also be used as a definition. ${\displaystyle \operatorname {pol} ({\mathcal {K}})=-\operatorname {dual} ({\mathcal {K}})}$

Self dual cone

A cone is called self-dual if true. ${\displaystyle \operatorname {dual} ({\mathcal {K}})={\mathcal {K}}}$

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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. ${\displaystyle \mathbb {R} ^{2}}$ ${\displaystyle {\mathcal {K}}:=\{x\in \mathbb {R} ^{2}\,|\,x_{1}\geq 0\,,x_{2}=0\}=\lambda (1,0)^{T}}$ ${\displaystyle \lambda \geq 0}$ ${\displaystyle \operatorname {dual} ({\mathcal {K}})=\mathbb {R} ^{+}\times \mathbb {R} }$ ${\displaystyle x\in {\mathcal {K}}}$ ${\displaystyle x^{T}y=\lambda y_{1}}$ ${\displaystyle \geq 0}$ ${\displaystyle \lambda \geq 0}$ ${\displaystyle y_{1}\geq 0}$

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 ${\displaystyle \mathbb {R} ^{2}}$ ${\displaystyle \langle x;y\rangle _{A}:=x^{T}Ay}$ ${\displaystyle A}$

${\displaystyle A={\begin{pmatrix}4&1\\1&2\end{pmatrix}}}$

is, so is the dual cone

${\displaystyle \operatorname {dual} ({\mathcal {K}})=\{y\in \mathbb {R} ^{2}\,|\,4y_{1}+y_{2}\geq 0\}}$.

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. ${\displaystyle y_{2}=-4y_{1}}$

An example of a self-dual cone is . ${\displaystyle {\mathcal {K}}:=\{x\in \mathbb {R} ^{2}\,|\,x_{1}\geq 0\,,x_{2}\geq 0\}}$

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 . ${\displaystyle V}$ ${\displaystyle V^{*}}$

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 .