Quaternion-Kähler symmetric space

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

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

The theorem has applications in optimization, where it sometimes is used to solve minimax problems.

Statement

The theorem applies to the following situation. 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. Further assume that $\varphi (x,z)$ is convex in $x$ for every $z\in Z$.

Under these conditions, Danskin's theorem provides conclusions regarding the differentiability of the function

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

To state these results, we define the set of maximizing points $Z_0(x)$ as

$Z_0(x)=\{\bar{z}:\varphi (x,\bar{z})={\max }_{z\in Z}\varphi (x,z)\}.$

Danskin's theorem then provides the following results.

Convexity

$f(x)$ is convex.

Directional derivatives

The directional derivative of $f(x)$ in the direction $y$, denoted $D_{y}f(x)$, is given by

$D_{y}f(x)={\max }_{z\in Z_0(x)}{\varphi }^{\prime }(x,z;y),$

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

$\frac{\partial f}{\partial x}=\frac{\partial \varphi (x,\bar{z})}{\partial x}.$

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.

See also

• Maximum theorem
• Envelope theorem
• Hotellings Lemma

References

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