Danskin's Theorem

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

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

## Statement

The theorem applies to the following situation. Suppose <math>phi(x,z)</math> is a continuous function of two arguments,

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

Danskin's theorem then provides the following results.

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- <math>f(x) = max_ phi(x,z).</math>

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

- <math>phi: ^n times Z rightarrow </math>

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

- <math>f(x) = max_ phi(x,z).</math>

- <math>Z_0(x) = left.</math>

Danskin's theorem then provides the following results.

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

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