# 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)}$

### 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 . ${\ displaystyle V}$

## 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. ${\ displaystyle \ operatorname {dual} ({\ mathcal {K}}) = {\ mathcal {K}}}$

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