Danskin's theorem

In convex analysis, Danskin's theorem is a theorem which provides information about the derivatives of a function of the form

$$f(x)=\max _{z\in Z}\phi (x,z).$$

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,

$$\phi :\mathbb {R} ^{n}\times Z\to \mathbb {R}$$

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

$$f(x)=\max _{z\in Z}\phi (x,z).$$

To state these results, we define the set of maximizing points $Z_{0}(x)$ as

$$Z_{0}(x)=\left\{{\overline {z}}:\phi (x,{\overline {z}})=\max _{z\in Z}\phi (x,z)\right\}.$$

Danskin's theorem then provides the following results.

Convexity

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

Directional semi-differential

The semi-differential of $f(x)$ in the direction $y$, denoted $\partial _{y}\ f(x),$ is given by

$$\partial _{y}f(x)=\max _{z\in Z_{0}(x)}\phi '(x,z;y),$$

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 ${\overline {z}}$. In this case, the derivative of $f(x)$ (or the gradient of $f(x)$ if $x$ is a vector) is given by

$${\frac {\partial f}{\partial x}}={\frac {\partial \phi (x,{\overline {z}})}{\partial x}}.$$

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 $Z=\{-1,+1\},\ \phi (x,z)=zx$, 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

$$\partial f(x)=\mathrm {conv} \left\{{\frac {\partial \phi (x,z)}{\partial x}}:z\in Z_{0}(x)\right\}$$

where $\mathrm {conv}$ 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 $\phi (\cdot ,z)$ 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 $\operatorname {int} (\operatorname {dom} (f)),$ the interior of the effective domain of $f,$ is nonempty, and that $\phi$ is continuous on the set $\operatorname {int} (\operatorname {dom} (f))\times Z.$ Then for all $x$ in $\operatorname {int} (\operatorname {dom} (f)),$ the subdifferential of $f$ at $x$ is given by

$$\partial f(x)=\operatorname {conv} \left\{\partial \phi (x,z):z\in Z_{0}(x)\right\}$$

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