generalized Farkas lemma

generalized Farkas lemma

Farkas’ Lemma

of convex optimization
and

linear programming

can be formulated
for

topological vector spaces

.

The more abstract version of Farkas’ Lemma is useful
for understanding the essence of the usual version
of the lemma proven for matrices,
and of course, for solving optimization problems
in infinite-dimensional spaces.

The key insight
is that the notion of linear inequalities in
a finite number of real variables
can be generalized to abstract

linear spaces

by the

concept

of a cone.

0.1

Formal statements

Farkas’ Lemma may be stated in the several

equivalent

ways.

Theorem

1 is conceptually the simplest,
but Theorem 2 and 3 are more convenient for applications.

Theorem 1

.

Let

X

be a

real vector spaces

, and

X

′

be a

subspace

of

linear
functionals

on

X

that separate points.
Impose on

X

the

weak topology

generated by

X

′

.

Given

x

∈

X

,
and a weakly-closed convex cone

K

⊆

X

,
the following are equivalent:

(a)

x

∈

K

.

(b)

If

ϕ

∈

X

′

satisfies

ϕ

⁢

(

y

)

≥

0

for all

y

∈

K

,
then

ϕ

⁢

(

x

)

≥

0

.

(c)

ϕ

⁢

(

x

)

≥

0

for all

ϕ

∈

K

+

(

anti-cone

of

K

with respect to

X

′

).

Proof.

The equivalence of conditions (a) and (c) is a fundamental property
of the anti-cone, while condition (b) is merely
a rephrasal of condition (c).
∎

Theorem 2 is a version of Theorem 1 where
the

vector space

and its

dual space

switch roles.

Theorem 2

.

Let

X

be a real vector space, and

X

′

be a subspace of linear functionals on

X

that separate points.
Impose on

X

′

the weak-* topology generated by

X

.

Given a

functional

f

∈

X

′

, and a weak-* closed convex cone

K

⊆

X

′

,
the following are equivalent:

(a)

f

∈

K

.

(b)

{

x

∈

X

:

f

⁢

(

x

)

≥

0

}

⊇

⋂

ϕ

∈

K

{

x

∈

X

:

ϕ

⁢

(

x

)

≥

0

}

.

Proof.

Make the substitutions

X

→

X

′

,

X

′

→

X

,

x

→

f

and

K

→

K

in Theorem 1.
∎

Theorem 3 incorporates

inequalities

defined
by

linear mappings

; such linear mappings are the analogues to the matrices
involved in the

finite-dimensional

version of Farkas’ Lemma.

Theorem 3

.

Let

X

and

Y

be real vector spaces,
with corresponding spaces of linear functionals

X

′

and

Y

′

that separate points.
Have

X

′

and

Y

′

generate the weak topology for

X

and

Y

respectively.

Given

y

∈

Y

,
a linear mapping

T

:

X

→

Y

, and
a subset

K

⊆

X

such that

T

⁢

(

K

)

is a weakly-closed convex cone,
the following are equivalent:

(a)

The

linear equation

T

⁢

x

=

y

has a solution

x

∈

K

.

(b)

If

ψ

∈

Y

′

satisfies

ψ

⁢

(

T

⁢

x

)

≥

0

for all

x

∈

K

,
then

ψ

⁢

(

y

)

≥

0

.

(c)

If

ψ

∈

Y

′

satisfies

T

*

⁢

ψ

∈

K

+

(anti-cone of

K

with respect to

X

′

),
then

ψ

⁢

(

y

)

≥

0

.

Here

T

*

:

Y

′

→

X

′

denotes the

pullback

, restricted to

Y

′

and

X

′

,
defined by

T

*

⁢

ψ

=

ψ

∘

T

.

Proof.

Make the substitutions

X

→

Y

,

X

′

→

Y

′

,

x

→

y

and

K

→

T

⁢

(

K

)

in Theorem 1.
Condition (c) is a rephrasal of condition (b).
∎

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 Kincaid and Ward Cheney.

Numerical Analysis: Mathematics of Scientific Computing

,
third edition. Brooks/Cole, 2002.

Title

generalized Farkas lemma

Canonical name

GeneralizedFarkasLemma

Date of creation

2013-03-22 17:20:53

Last modified on

2013-03-22 17:20:53

Owner

stevecheng (10074)

Last modified by

stevecheng (10074)

Numerical id

7

Author

stevecheng (10074)

Entry type

Theorem

Classification

msc 46A03

Classification

msc 46A20

Classification

msc 15A39

Classification

msc 49J35

Synonym

Farkas lemma for topological vector spaces

Synonym

generalized Farkas theorem

Related topic

AntiCone

Related topic

Cone5

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