Danskin's theorem

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. The original theorem given by J. M. Danskin in his 1967 monograph provides a formula for the directional derivative of the maximum of a directionally differentiable function.
An extension to more general conditions was proven 1971 by Dimitri Bertsekas.

Statement

The following version is proven in "Nonlinear programming". Suppose is a continuous function of two arguments,
where is a compact set.
Under these conditions, Danskin's theorem provides conclusions regarding the convexity and differentiability of the function
To state these results, we define the set of maximizing points as
Danskin's theorem then provides the following results.
;Convexity
;Directional semi-differential
;Derivative

Example of no [directional derivative]

In the statement of Danskin, it is important to conclude semi-differentiability of and not directional-derivative as explains this simple example.
Set, we get which is semi-differentiable with but has not a directional derivative at.

Extension

The 1971 Ph.D. Thesis by Dimitri P. Bertsekas proves a more general result, which does not require that is differentiable. Instead it assumes that is an extended real-valued closed proper convex function for each in the compact set that the interior of the effective domain of is nonempty, and that is continuous on the set Then for all in the subdifferential of at is given by
where is the subdifferential of at for any in