Danskin's theorem

In convex analysis, Danskin's theorem is a theorem which provides information about the derivatives of a function of the form
$f(x) = \max_ \phi(x,z).$

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 (not necessarily convex) directionally differentiable function.

An extension to more general conditions was proven 1971 by Dimitri Bertsekas.

Statement

The following version is proven in "Nonlinear programming" (1991). Suppose $\phi(x,z)$ is a continuous function of two arguments,
$\phi : \R^n \times Z \to \R$
where $Z \subset \R^m$ is a compact set. Further assume that $\phi(x,z)$ is convex in $x$ for every $z \in Z.$

Under these conditions, Danskin's theorem provides conclusions regarding the convexity and differentiability of the function
$f(x) = \max_ \phi(x,z).$
To state these results, we define the set of maximizing points $Z_0(x)$ as
$Z_0(x) = \left\.$

Danskin's theorem then provides the following results.

;Convexity
: $f(x)$ is convex.

;Directional semi-differential
: The semi-differential of $f(x)$ in the direction $y$, denoted $\partial_y\ f(x),$ is given by $\partial_y f(x) = \max_ \phi'(x,z;y),$ where $\phi'(x,z;y)$ is the directional derivative of the function $\phi(\cdot,z)$ at $x$ in the direction $y.$

;Derivative
: $f(x)$ is differentiable at $x$ if $Z_0(x)$ consists of a single element $\overline$. In this case, the derivative of $f(x)$ (or the gradient of $f(x)$ if $x$ is a vector) is given by $\frac = \frac.$

Example of no

directional derivative

In the statement of Danskin, it is important to conclude semi-differentiability of $f$ and not directional-derivative as explains this simple example.
Set $Z=, \phi(x,z)= zx,$, we get $f(x)=, x,$ which is semi-differentiable with $\partial_-f(0)=-1, \partial_+f(0)=+1$ but has not a directional derivative at $x=0$.

Subdifferential

:If $\phi(x,z)$ is differentiable with respect to $x$ for all $z \in Z,$ and if $\partial \phi/\partial x$ is continuous with respect to $z$ for all $x$, then the subdifferential of $f(x)$ is given by $\partial f(x) = \mathrm \left\$ where $\mathrm$ indicates the convex hull operation.

Extension

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

proves a more general result, which does not require that $\phi(\cdot,z)$ is differentiable. Instead it assumes that $\phi(\cdot,z)$ is an extended real-valued closed proper convex function for each $z$ in the compact set $Z,$ that $\operatorname(\operatorname(f)),$ the interior of the effective domain of $f,$ is nonempty, and that $\phi$ is continuous on the set $\operatorname(\operatorname(f)) \times Z.$ Then for all $x$ in $\operatorname(\operatorname(f)),$ the subdifferential of $f$ at $x$ is given by
$\partial f(x) = \operatorname \left\$
where $\partial \phi(x,z)$ is the subdifferential of $\phi(\cdot,z)$ at $x$ for any $z$ in $Z.$

See also

* Maximum theorem
* Envelope theorem
* Hotelling's lemma

References

{{Convex analysis and variational analysis

Convex optimization
Theorems in analysis