tests for local extrema in Lagrange multiplier method

tests for local extrema in Lagrange multiplier method

Let

U

be open in

ℝ

n

,
and

f

:

U

→

ℝ

,

g

:

U

→

ℝ

m

be twice

continuously differentiable functions

.
Assume that

p

∈

U

is a

stationary point

for

f

on

M

=

g

-

1

⁢

(

{

0

}

)

,
and

D

⁡

g

has full rank everywhere

1

1

Actually, only

D

⁡

g

⁢

(

p

)

needs to have full rank, and the

arguments

presented here continue to hold in that case, although

M

would not necessarily be a manifold then.

on

M

.
Then we know that

p

is the solution
to the

Lagrange multiplier

system

D

⁡

f

⁢

(

p

)

=

λ

⋅

D

⁡

g

⁢

(

p

)

,

(1)

for a Lagrange multiplier vector

λ

=

(

λ

1

,

…

,

λ

m

)

.

Our aim is to develop an analogue of the

second derivative test

for the stationary point

p

.

The most straightforward way to proceed is to consider a coordinate chart

α

:

V

→

M

for the manifold

M

, and consider the

Hessian

of the

function

f

∘

α

:

V

→

ℝ

n

at

0

=

α

-

1

⁢

(

p

)

. This Hessian is in fact just the

Hessian form

of

f

:

M

→

ℝ

expressed in the

coordinates

of the chart

α

. But the whole point of using Lagrange multipliers
is to avoid calculating coordinate charts directly, so we find an

equivalent

expression

for

D

2

⁡

(

f

∘

α

)

⁢

(

0

)

in terms of

D

2

⁡

f

⁢

(

p

)

without mentioning

derivatives

of

α

.

To do this, we

differentiate

f

∘

α

twice using the

chain rule

and

product
rule

2

2

Note that the “

product

”

operation

involved (second equality of (

2

))
is the operation of

composition of two

linear mappings

.
Think hard about this if you are not sure; it took me several tries to get this

formula

right,
since multi-variable iterated derivatives have a complicated

structure

.

.
To reduce clutter, from now on we use the prime notation for derivatives rather than

D

.

(

f

∘

α

)

′′

⁢

(

0

)

=

(

(

f

′

∘

α

)

⋅

α

′

)

′

⁢

(

0

)

=

(

(

f

′′

∘

α

)

⋅

α

′

⋅

α

′

+

(

f

′

∘

α

)

⋅

α

′′

)

⁢

(

0

)

=

f

′′

⁢

(

α

⁢

(

0

)

)

⋅

α

′

⁢

(

0

)

⋅

α

′

⁢

(

0

)

+

f

′

⁢

(

α

⁢

(

0

)

)

⋅

α

′′

⁢

(

0

)

=

f

′′

⁢

(

p

)

⋅

α

′

⁢

(

0

)

⋅

α

′

⁢

(

0

)

+

f

′

⁢

(

p

)

⋅

α

′′

⁢

(

0

)

.

(2)

If we interpret

(

f

∘

α

)

′′

⁢

(

0

)

as a bilinear mapping of vectors

u

,

v

∈

ℝ

n

-

m

,
then formula (

2

) really means

(

f

∘

α

)

′′

⁢

(

0

)

⋅

(

u

,

v

)

=

f

′′

⁢

(

p

)

⋅

(

α

′

⁢

(

0

)

⋅

u

,

α

′

⁢

(

0

)

⋅

v

)

+

f

′

⁢

(

p

)

⋅

(

α

′′

⁢

(

0

)

⋅

(

u

,

v

)

)

.

(3)

To obtain the

quadratic form

, we set

v

=

u

; also we abbreviate the vector

α

′

⁢

(

0

)

⋅

u

by

h

,
which belongs to the

tangent space

T

p

⁢

M

of

M

at

p

. So,

(

f

∘

α

)

′′

⁢

(

0

)

⋅

u

2

=

f

′′

⁢

(

p

)

⋅

h

2

+

f

′

⁢

(

p

)

⋅

(

α

′′

⁢

(

0

)

⋅

u

2

)

.

(4)

Naïvely, we might think that

(

f

∘

α

)

′′

⁢

(

0

)

is simply

f

′′

⁢

(

p

)

restricted to the tangent space

T

p

⁢

M

.
This happens to be the first term in (

4

), but there is also
an additional contribution by the second term involving

α

′′

⁢

(

0

)

;
intuitively,

α

′′

⁢

(

0

)

is the

curvature

of the surface (manifold)

M

,
“changing the

geometry

” of the graph of

f

.

But the second term of (

4

) still involves

α

.
To eliminate it, we differentiate the equation

g

∘

α

=

0

twice.

0

=

(

g

∘

α

)

′′

⁢

(

0

)

=

g

′′

⁢

(

p

)

⋅

α

′

⁢

(

0

)

⋅

α

′

⁢

(

0

)

+

g

′

⁢

(

p

)

⋅

α

′′

⁢

(

0

)

.

(5)

(It is derived the same way as (

2

) but with

f

replaced by

g

.)
Now we can substitute (

5

) and (

1

)
in (

2

) to eliminate the term

f

′

⁢

(

p

)

⋅

α

′′

⁢

(

0

)

:

(

f

∘

α

)

′′

⁢

(

0

)

=

f

′′

⁢

(

p

)

⋅

α

′

⁢

(

0

)

⋅

α

′

⁢

(

0

)

+

λ

⋅

g

′

⁢

(

p

)

⋅

α

′′

⁢

(

0

)

=

f

′′

⁢

(

p

)

⋅

α

′

⁢

(

0

)

⋅

α

′

⁢

(

0

)

-

λ

⋅

g

′′

⁢

(

p

)

⋅

α

′

⁢

(

0

)

⋅

α

′

⁢

(

0

)

,

(6)

or expressed as a quadratic form,

(

f

∘

α

)

′′

⁢

(

0

)

⋅

u

2

=

f

′′

⁢

(

p

)

⋅

h

2

-

λ

⋅

g

′′

⁢

(

p

)

⋅

h

2

.

(7)

Thus, to understand the nature of the stationary point

p

, we can study the modified Hessian:

f

′′

⁢

(

p

)

-

λ

⋅

g

′′

⁢

(

p

)

,

restricted to

T

p

⁢

M

.

(8)

For example, if this

bilinear form

is

positive definite

, then

p

is a

local minimum

,
and if it is

negative definite

, then

p

is a

local maximum

, and so on.
All the tests that apply to the usual Hessian in

ℝ

n

apply to the modified Hessian (

8

).

In coordinates of

ℝ

n

, the modified Hessian (

8

)
takes the form

∑

i

=

1

n

∑

j

=

1

n

(

∂

2

⁡

f

∂

⁡

x

i

⁢

∂

⁡

x

j

|

p

-

∑

k

=

1

m

λ

k

⁢

∂

2

⁡

g

k

∂

⁡

x

i

⁢

∂

⁡

x

j

|

p

)

⁢

h

i

⁢

h

j

.

(9)

We emphasize that the vector

h

can be restricted to lie in the tangent space

T

p

⁢

M

,
when studying the stationary point

p

of

f

restricted to

M

.

In matrix form (

9

)
can be written

B

=

[

∂

2

⁡

f

∂

⁡

x

i

⁢

∂

⁡

x

j

-

∑

k

=

1

m

λ

k

⁢

∂

2

⁡

g

k

∂

⁡

x

i

⁢

∂

⁡

x

j

]

i

⁢

j

.

(10)

But again, the test vector

h

need only lie on

T

p

⁢

M

,
so if we want to apply

positive

/negative definiteness tests for matrices,
they should instead be applied to the

projected

or

reduced

Hessian:

Z

t

⁢

B

⁢

Z

(11)

where the columns of the

n

×

(

n

-

m

)

matrix

Z

form a

basis

for

T

p

⁢

M

=

ker

⁡

g

′

⁢

(

p

)

⊂

ℝ

n

.

Title

tests for local extrema in Lagrange multiplier method

Canonical

name

TestsForLocalExtremaInLagrangeMultiplierMethod

Date of creation

2013-03-22 15:28:52

Last modified on

2013-03-22 15:28:52

Owner

stevecheng (10074)

Last modified by

stevecheng (10074)

Numerical id

6

Author

stevecheng (10074)

Entry type

Result

Classification

msc 26B12

Classification

msc 49-00

Classification

msc 49K35

Related topic

HessianForm

Related topic

RelationsBetweenHessianMatrixAndLocalExtrema

Defines

projected Hessian

Defines

reduced Hessian

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