anti-cone

anti-cone

Let

X

be a

real vector space

, and

Φ

be a

subspace

of

linear functionals

on

X

.

For any set

S

⊆

X

,
its

anti-cone

S

+

,
with respect to

Φ

, is the set

S

+

=

{

ϕ

∈

Φ

:

ϕ

⁢

(

x

)

≥

0

,

for all

⁢

x

∈

S

}

.

The anti-cone is also called the

dual cone

.

Usage

The anti-cone

operation

is generally applied to subsets of

X

that are themselves
cones.
Recall that a cone in a real vector space generalize the notion of
linear inequalities in a finite number of real variables.
The dual cone provides a natural way to transfer such

inequalities

in the original

vector space

to its

dual space

.
The

concept

is useful in the theory of
duality.

The set

Φ

in the

definition

may be taken to be any subspace
of the algebraic dual space

X

*

.
The set

Φ

often needs to be restricted
to a subspace smaller than

X

*

, or even
the continuous dual space

X

′

,
in order to obtain
the nice

closure

and

reflexivity

properties

below.

Basic properties

Property 1

.

The anti-cone is a convex cone in

Φ

.

Proof.

If

ϕ

⁢

(

x

)

is non-negative, then so is

t

⁢

ϕ

⁢

(

x

)

for

t

>

0

.
And if

ϕ

1

⁢

(

x

)

,

ϕ

2

⁢

(

x

)

≥

0

,
then clearly

(

1

-

t

)

⁢

ϕ

1

⁢

(

x

)

+

t

⁢

ϕ

2

⁢

(

x

)

≥

0

for

0

≤

t

≤

1

.
∎

Property 2

.

If

K

⊆

X

is a cone, then its anti-cone

K

+

may be equivalently
characterized as:

K

+

=

{

ϕ

∈

Φ

:

ϕ

⁢

(

x

)

over

x

∈

K

is bounded below

}

.

Proof.

It suffices to show that if

inf

x

∈

K

⁡

ϕ

⁢

(

x

)

is bounded below,
then it is non-negative.
If it were

negative

, take some

x

∈

K

such that

ϕ

⁢

(

x

)

<

0

. For any

t

>

0

, the vector

t

⁢

x

is in the cone

K

,
and the function value

ϕ

⁢

(

t

⁢

x

)

=

t

⁢

ϕ

⁢

(

x

)

would be arbitrarily
large negative, and hence

unbounded

below.
∎

Topological properties

Assumptions

.

Assume that

Φ

separates points of

X

.
Let

X

have the

weak topology

generated by

Φ

,
and let

Φ

have the weak-*

topology

generated by

X

;
this makes

X

and

Φ

into Hausdorff

topological vector spaces

.

Vectors

x

∈

X

will be identified
with their images

x

^

under the natural embedding of

X

in its double dual space.

The

pairing

(

X

,

Φ

)

is sometimes called a dual pair;
and

(

Φ

,

X

)

, where

X

is identified with its image in the double dual,
is also a dual pair.

Property 3

.

S

+

is weak-* closed.

Proof.

Let

{

ϕ

α

}

⊆

Φ

be a net converging to

ϕ

in the weak-* topology.
By definition,

x

^

⁢

(

ϕ

α

)

=

ϕ

α

⁢

(

x

)

≥

0

.
As the

functional

x

^

is

continuous

in the weak-* topology,
we have

x

^

⁢

(

ϕ

α

)

→

x

^

⁢

(

ϕ

)

≥

0

.
Hence

ϕ

∈

S

+

.
∎

Property 4

.

S

¯

+

=

S

+

.

Proof.

The

inclusion

S

¯

+

⊆

S

+

is

obvious

.
And if

ϕ

⁢

(

x

)

≥

0

for all

x

∈

S

,
then by

continuity

, this holds true for

x

∈

S

¯

too —
so

S

¯

+

⊇

S

+

.
∎

Properties involving cone inclusion

Property 5

(Farkas’ lemma).

Let

K

⊆

X

be a weakly-closed convex cone.
Then

x

∈

K

if and only if

ϕ

⁢

(

x

)

≥

0

for all

ϕ

∈

K

+

.

Proof.

That

ϕ

⁢

(

x

)

≥

0

for

ϕ

∈

K

+

and

x

∈

K

is just the definition.
For the

converse

, we show that if

x

∈

X

∖

K

,
then there exists

ϕ

∈

K

+

such that

ϕ

⁢

(

x

)

<

0

.

If

K

=

∅

, then the desired

ϕ

∈

Φ

=

K

+

exists because

Φ

can separate the points

x

and

0

.
If

K

≠

∅

, by the hyperplane separation theorem,
there is a

ϕ

∈

Φ

such that

ϕ

⁢

(

x

)

<

inf

y

∈

K

⁡

ϕ

⁢

(

y

)

.
This

ϕ

will automatically be in

K

+

by Property

2

.
The

zero vector

is the weak limit of

t

⁢

y

, as

t

↘

0

,
for any vector

y

.
Thus

0

∈

K

, and
we conclude with

inf

y

∈

K

⁡

ϕ

⁢

(

y

)

≤

0

.
∎

Property 6

.

K

+

+

=

K

¯

for any convex cone

K

.
(The anti-cone operation on

K

+

is to be taken with respect to

X

.)

Proof.

We work with

K

¯

, which is a weakly-closed convex cone.
By Property

5

,

x

∈

K

¯

if and only if

ϕ

⁢

(

x

)

≥

0

for all

ϕ

∈

K

¯

+

=

K

+

.
But by definition of the second anti-cone,

x

^

∈

(

K

+

)

+

if and only if

ϕ

⁢

(

x

)

=

x

^

⁢

(

ϕ

)

≥

0

for all

ϕ

∈

K

+

.
∎

Property 7

.

Let

K

and

L

be convex cones in

X

, with

K

weakly closed.
Then

K

+

⊆

L

+

if and only if

K

⊇

L

.

Proof.

K

+

⊆

L

+

⟹

K

=

K

¯

=

K

+

+

⊇

L

+

+

=

L

¯

⊇

L

⟹

K

+

⊆

L

+

.

∎

References

1

B. D. Craven and J. J. Kohila.
“

Generalizations

of Farkas’

Theorem

.”

SIAM Journal on

Mathematical Analysis

.
Vol. 8, No. 6, November 1977.

2

David G. Luenberger.

Optimization by Vector Space Methods

.
John Wiley & Sons, 1969.

Title

anti-cone

Canonical

name

Anticone

Date of creation

2013-03-22 17:20:48

Last modified on

2013-03-22 17:20:48

Owner

stevecheng (10074)

Last modified by

stevecheng (10074)

Numerical id

8

Author

stevecheng (10074)

Entry type

Definition

Classification

msc 46A03

Classification

msc 46A20

Synonym

anticone

Synonym

dual cone

Related topic

GeneralizedFarkasLemma

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