# Danskin's theorem

to get instant updates about 'Danskin's Theorem' on your MyPage. Meet other similar minded people. Its Free!

X

Description:
In convex analysis, Danskin's theorem is a theorem which provides information about the derivatives of a function of the form

[itex]f(x) = max_ phi(x,z).[/itex]

The theorem has applications in optimization, where it sometimes is used to solve minimax problems.

## Statement

The theorem applies to the following situation. Suppose [itex]phi(x,z)[/itex] is a continuous function of two arguments,
[itex]phi: ^n times Z rightarrow [/itex]
where [itex]Z subset ^m[/itex] is a compact set. Further assume that [itex]phi(x,z)[/itex] is convex in [itex]x[/itex] for every [itex]z in Z[/itex].

Under these conditions, Danskin's theorem provides conclusions regarding the differentiability of the function
[itex]f(x) = max_ phi(x,z).[/itex]
To state these results, we define the set of maximizing points [itex]Z_0(x)[/itex] as
[itex]Z_0(x) = left.[/itex]

Danskin's theorem then provides the following results.

Convexity
[itex]f(x)[/itex] is convex.
Directional derivatives
The directional derivative of [itex]f(x)[/itex] in the direction [itex]y[/itex], denoted [itex]D_y f(x)[/itex], is given by
:[itex]D_y f(x) = max_ phi'(x,z;y),[/itex]
where [itex]phi'(x,z;y)[/itex] is the directional derivative of the function [itex]phi(cdot,z)[/itex] at [itex]x[/itex] in the......
...

No feeds found

All