Danskins Theorem

f(x) = \max_{z \in Z} \phi(x,z).

The theorem has applications in Optimization , where it sometimes is used to solve Minimax problems.

STATEMENT

The theorem applies to the following situation. Suppose

\phi(x,z)

is a Continuous Function of two arguments,
:

\phi: {\mathbb R}^n 	imes Z 
ightarrow {\mathbb R}

where

Z \subset {\mathbb R}^m

is a Compact Set . Further assume that

\phi(x,z)

is Convex in

x

for every

z \in Z

.

Under these conditions, Danskin's theorem provides conclusions regarding the Differentiability of the function
:

f(x) = \max_{z \in Z} \phi(x,z).

To state these results, we define the set of maximizing points

Z_0(x)

as
:

Z_0(x) = \left\{ \overline{z} : \phi(x,\overline{z}) = \max_{z \in Z} \phi(x,z)
ight\}.

Danskin's theorem then provides the following results.

;Convexity
:

f(x)

is Convex .
;Directional derivatives
: The Directional Derivative of

f(x)

in the direction

y

, denoted

D_y\ f(x)

, is given by
::

D_y f(x) = \max_{z \in Z_0(x)} \phi(x,z).

;Derivative
:

f(x)

is Differentiable at

x

Z_0(x)

consists of a single element

\overline{z}

. In this case, the Derivative of

f(x)

(or the Gradient of

f(x)

x

is a vector) is given by
::

rac{\partial f}{\partial x} = rac{\partial \phi(x,\overline{z})}{\partial x}.

;Subdifferential
:If

\phi(x,z)

is differentiable with respect to

x

for all

z \in Z

, and if

\partial \phi/\partial x

is continuous with respect to

z

for all

x

, then the Subdifferential of

f(x)

is given by
::

\partial f(x) = \mathrm{conv} \left\{ rac{\partial \phi(x,z)}{\partial x} : z \in Z_0(x) 
ight\}

: where

\mathrm{conv}

indicates the Convex Hull operation.

REFERENCES

	CATEGORIES ABOUT DANSKINS THEOREM

	convex analysis

	optimization

	mathematical theorems

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Information About

Danskins Theorem