convex function

convex function

Definition

Suppose

Ω

is a

convex set

in a

vector space

over

ℝ

(or

ℂ

), and suppose

f

is a function

f

:

Ω

→

ℝ

.
If for any

x

,

y

∈

Ω

,

x

≠

y

and any

λ

∈

(

0

,

1

)

, we have

f

⁢

(

λ

⁢

x

+

(

1

-

λ

)

⁢

y

)

≤

λ

⁢

f

⁢

(

x

)

+

(

1

-

λ

)

⁢

f

⁢

(

y

)

,

we say that

f

is a

convex function

.
If for any

x

,

y

∈

Ω

and any

λ

∈

(

0

,

1

)

, we have

f

⁢

(

λ

⁢

x

+

(

1

-

λ

)

⁢

y

)

≥

λ

⁢

f

⁢

(

x

)

+

(

1

-

λ

)

⁢

f

⁢

(

y

)

,

we say that

f

is a

concave function

. If either of the

inequalities

are strict, then we say that

f

is a

strictly convex function

,
or a

strictly concave function

, respectively.

Properties

•

A function

f

is a (strictly) convex function if and only if

-

f

is
a (strictly) concave function. For this reason, most of the below discussion only
focuses on convex functions. Analogous result holds for concave functions.

•

On

ℝ

, a

continuous function

is convex
if and only if for all

x

,

y

∈

ℝ

, we have

f

⁢

(

x

+

y

2

)

≤

f

⁢

(

x

)

+

f

⁢

(

y

)

2

.

•

On

ℝ

, a once

differentiable function

is convex if and only if

f

′

is

monotone increasing

.

•

Suppose

f

is twice

continuously differentiable function

on

ℝ

.
Then

f

is convex if and only if

f

′′

≥

0

.
If

f

′′

>

0

, then

f

is strictly convex.

•

A

local minimum

of a convex function is a

global minimum

.
See this page (

http://planetmath.org/ExtremalValueOfConvexconcaveFunctions

).

Examples

•

e

x

,

e

-

x

, and

x

2

are convex functions on

ℝ

. Also,

x

4

is strictly
convex, but

12

⁢

x

2

vanishes

at

x

=

0

.

•

A norm (

http://planetmath.org/NormedVectorSpace

) is a convex function.

Remark.

We may generalize the above definition of a convex function to an that of an extended real-valued function whose domain is not necessarily a convex set. First, we define what an

epigraph

of a function is.

Let

Ω

be a subset of a vector space over the reals, and

f

an extended real-valued function defined on

Ω

. The

epigraph

of

f

, denoted by

epi

⁡

(

f

)

, is the set

{

(

x

,

r

)

∣

x

∈

Ω

,

⁢

r

≥

f

⁢

(

x

)

}

.

An extended real-valued function

f

defined on a subset

Ω

of a vector space

V

over the reals is said to be

convex

if its

epigraph

is a

convex subset

of

V

×

ℝ

. With this definition, the domain

Ω

of

f

need not be convex. However, its subset

{

x

∈

Ω

∣

f

⁢

(

x

)

<

∞

}

, called the

effective domain

and denoted by

eff

.

dom

⁡

(

f

)

, is convex. To see this, suppose

x

,

y

∈

eff

.

dom

⁡

(

f

)

and

z

=

λ

⁢

x

+

(

1

-

λ

)

⁢

y

with

0

≤

λ

≤

0

. Then

(

z

,

z

¯

)

=

λ

⁢

(

x

,

f

⁢

(

x

)

)

+

(

1

-

λ

)

⁢

(

y

,

f

⁢

(

y

)

)

∈

epi

⁡

(

f

)

, where

z

¯

=

λ

⁢

f

⁢

(

x

)

+

(

1

-

λ

)

⁢

f

⁢

(

y

)

, since

epi

⁡

(

f

)

is convex by definition. Therefore,

z

∈

dom

⁡

(

f

)

. In fact,

f

⁢

(

z

)

≤

z

¯

=

λ

⁢

f

⁢

(

x

)

+

(

1

-

λ

)

⁢

f

⁢

(

y

)

<

∞

, which implies that

z

∈

eff

.

dom

⁡

(

f

)

.

Title

convex function

Canonical name

ConvexFunction

Date of creation

2013-03-22 11:46:26

Last modified on

2013-03-22 11:46:26

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

28

Author

matte (1858)

Entry type

Definition

Classification

msc 52A41

Classification

msc 26A51

Classification

msc 26B25

Classification

msc 55Q05

Classification

msc 18G30

Classification

msc 18B40

Classification

msc 20J05

Classification

msc 20E07

Classification

msc 18-01

Classification

msc 20L05

Related topic

JensensInequality

Related topic

LogarithmicallyConvexFunction

Defines

concave function

Defines

strictly convex function

Defines

strictly concave function

Defines

strictly convex

Defines

strictly concave

Defines

epigraph

Defines

effective domain

Defines

concave

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