Danskin's theorem

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

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

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,

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

where <math>Z subset ^m</math> is a compact set. Further assume that <math>phi(x,z)</math> is convex in <math>x</math> for every <math>z in Z</math>.

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

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

To state these results, we define the set of maximizing points <math>Z_0(x)</math> as

<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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