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 by J. M. Danskin, given in his 1967, monograph "The Theory of Max-Min and its Applications to Weapons Allocation Problems," Springer, NY, provides a formula for the directional derivative of the maximum of a directionally differentiable function. When adapted to the case of a convex function, this formula yields the following theorem given in somewhat more general form as Proposition A.22 in the 1971 Ph.D. Thesis by D. P. Bertsekas, "Control of Uncertain Systems with a Set-Membership Description of the Uncertainty". A proof of the following version can be found in the 1999 book "Nonlinear Programming" by Bertsekas.

Statement

The theorem applies to the following situation. Suppose is a continuous function of two arguments,
where is a compact set. Further assume that is convex in for every.
Under these conditions, Danskin's theorem provides conclusions regarding the 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 derivatives
;Derivative
;Subdifferential
;Extension
The 1971 Ph.D. Thesis by 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.