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. 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.

Statement[edit]

The following version is proven in "Nonlinear programming" (1991).[2] 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 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
is convex.
Directional derivatives
The directional derivative of in the direction , denoted is given by
where is the directional derivative of the function at in the direction
Derivative
is differentiable at if consists of a single element . In this case, the derivative of (or the gradient of if is a vector) is given by
Subdifferential
If is differentiable with respect to for all and if is continuous with respect to for all , then the subdifferential of is given by
where indicates the convex hull operation.

Extension[edit]

The 1971 Ph.D. Thesis by Bertsekas (Proposition A.22) [3] 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

See also[edit]

References[edit]

  1. ^ Danskin, John M. (1967). The theory of Max-Min and its application to weapons allocation problems. New York: Springer.
  2. ^ Bertsekas, Dimitri P. (1999). Nonlinear programming (Second ed.). Belmont, Massachusetts. ISBN 1-886529-00-0.
  3. ^ Bertsekas, Dimitri P. (1971). Control of Uncertain Systems with a Set-Membership Description of Uncertainty (PDF) (PhD). Cambridge, MA: MIT.