Danskin's theorem

In convex analysis, Danskin's theorem is a theorem which provides information about the derivatives of a function of the form $$f(x)=\underset{z\in Z}{\max }\varphi (x,z).$$ 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 $\varphi (x,z)$ is a continuous function of two arguments, $$\varphi :{\mathbb{ℝ}}^n\times Z\to \mathbb{ℝ}$$ where $Z\subset {\mathbb{ℝ}}^m$ is a compact set. Under these conditions, Danskin's theorem provides conclusions regarding the convexity and differentiability of the function $$f(x)=\underset{z\in Z}{\max }\varphi (x,z).$$ To state these results, we define the set of maximizing points $Z_0(x)$ as $$Z_0(x)=\{\bar{z}:\varphi (x,\bar{z})=\underset{z\in Z}{\max }\varphi (x,z)\}.$$ Danskin's theorem then provides the following results.

Convexity

$f(x)$ is convex if $\varphi (x,z)$ is convex in $x$ for every $z\in Z$.

Directional semi-differential

The semi-differential of $f(x)$ in the direction $y$, denoted $$\begin{gathered} {\partial }_y \\ f(x), \end{gathered}$$ is given by $${\partial }_{y}f(x)=\underset{z\in Z_0(x)}{\max }{\varphi }^{\prime }(x,z;y),$$ where ${\varphi }^{\prime }(x,z;y)$ is the directional derivative of the function $\varphi (\cdot ,z)$ at $x$ in the direction $y.$

Derivative

$f(x)$ is differentiable at $x$ if $Z_0(x)$ consists of a single element $\bar{z}$. In this case, the derivative of $f(x)$ (or the gradient of $f(x)$ if $x$ is a vector) is given by $$\frac{\partial f}{\partial x}=\frac{\partial \varphi (x,\bar{z})}{\partial x}.$$

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 $$\begin{gathered} Z=\{-1,+1\}, \\ \varphi (x,z)=zx \end{gathered}$$, 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 $\varphi (x,z)$ is differentiable with respect to $x$ for all $z\in Z,$ and if $\partial \varphi /\partial x$ is continuous with respect to $z$ for all $x$, then the subdifferential of $f(x)$ is given by $$\partial f(x)=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{v}\{\frac{\partial \varphi (x,z)}{\partial x}:z\in Z_0(x)\}$$ where $\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{v}$ indicates the convex hull operation.

Extension

The 1971 Ph.D. Thesis by Dimitri P. Bertsekas (Proposition A.22) proves a more general result, which does not require that $\varphi (\cdot ,z)$ is differentiable. Instead it assumes that $\varphi (\cdot ,z)$ is an extended real-valued closed proper convex function for each $z$ in the compact set $Z,$ that $\mathrm{int}(\mathrm{dom}(f)),$ the interior of the effective domain of $f,$ is nonempty, and that $\varphi$ is continuous on the set $\mathrm{int}(\mathrm{dom}(f))\times Z.$ Then for all $x$ in $\mathrm{int}(\mathrm{dom}(f)),$ the subdifferential of $f$ at $x$ is given by $$\partial f(x)=\mathrm{conv}\{\partial \varphi (x,z):z\in Z_0(x)\}$$ where $\partial \varphi (x,z)$ is the subdifferential of $\varphi (\cdot ,z)$ at $x$ for any $z$ in $Z.$

See also

• Maximum theorem
• Envelope theorem
• Hotelling's lemma