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 [1] provides a formula for the directional derivative of the maximum of a (not necessarily convex) directionally differentiable function.
An extension to more general conditions was proven 1971 by Dimitri Bertsekas.
The following version is proven in "Nonlinear programming" (1991).
[2] Suppose
is a continuous function of two arguments,
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
Danskin's theorem then provides the following results.
In the statement of Danskin, it is important to conclude semi-differentiability of
and not directional-derivative as explains this simple example.
Set
but has not a directional derivative at
The 1971 Ph.D. Thesis by Dimitri P. Bertsekas (Proposition A.22) [3] proves a more general result, which does not require that
is an extended real-valued closed proper convex function for each
in the compact set
int ( dom ( f ) ) ,
{\displaystyle \operatorname {int} (\operatorname {dom} (f)),}
the interior of the effective domain of
is continuous on the set
int ( dom ( f ) ) ×
{\displaystyle \operatorname {int} (\operatorname {dom} (f))\times Z.}
int ( dom ( f ) ) ,
{\displaystyle \operatorname {int} (\operatorname {dom} (f)),}
∂ f ( x ) = conv