Danskin's theorem

The theorem applies to the following situation. Suppose ${\displaystyle \phi (x,z)}$ is a continuous function of two arguments,

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

where ${\displaystyle Z\subset {\mathbb {R} }^{m}}$ is a compact set. Further assume that ${\displaystyle \phi (x,z)}$ is convex in ${\displaystyle x}$ for every ${\displaystyle z\in Z}$.

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

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

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

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

Danskin's theorem then provides the following results.

Convexity

${\displaystyle f(x)}$ is convex.

Directional derivatives

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

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

where ${\displaystyle \phi '(x,z;y)}$ is the directional derivative of the function ${\displaystyle \phi (\cdot ,z)}$ at ${\displaystyle x}$ in the direction ${\displaystyle y}$.

Derivative

${\displaystyle f(x)}$ is differentiable at ${\displaystyle x}$ if ${\displaystyle Z_{0}(x)}$ consists of a single element ${\displaystyle {\overline {z}}}$. In this case, the derivative of ${\displaystyle f(x)}$ (or the gradient of ${\displaystyle f(x)}$ if ${\displaystyle x}$ is a vector) is given by

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

Subdifferential

If ${\displaystyle \phi (x,z)}$ is differentiable with respect to ${\displaystyle x}$ for all ${\displaystyle z\in Z}$, and if ${\displaystyle \partial \phi /\partial x}$ is continuous with respect to ${\displaystyle z}$ for all ${\displaystyle x}$, then the subdifferential of ${\displaystyle f(x)}$ is given by

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

where ${\displaystyle \mathrm {conv} }$ indicates the convex hull operation.

Extension

The 1971 Ph.D. Thesis by Bertsekas [1] (Proposition A.22) proves a more general result, which does not require that ${\displaystyle \phi (\cdot ,z)}$ is differentiable. Instead it assumes that ${\displaystyle \phi (\cdot ,z)}$ is an extended real-valued closed proper convex function for each ${\displaystyle z}$ in the compact set ${\displaystyle Z}$, that ${\displaystyle int(dom(f))}$, the interior of the effective domain of ${\displaystyle f}$, is nonempty, and that ${\displaystyle \phi }$ is continuous on the set ${\displaystyle int(dom(f))\times Z}$. Then for all ${\displaystyle x}$ in ${\displaystyle int(dom(f))}$, the subdifferential of ${\displaystyle f}$ at ${\displaystyle x}$ is given by

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

where ${\displaystyle \partial \phi (x,z)}$ is the subdifferential of ${\displaystyle \phi (\cdot ,z)}$ at ${\displaystyle x}$ for any ${\displaystyle z}$ in ${\displaystyle Z}$.

• Maximum theorem
• Envelope theorem
• Hotellings Lemma

• Danskin, John M. (1967). The Theory of Max-Min and its Applications to Weapons Allocation Problems. NY: Springer.
• Bertsekas, Dimitri P. (1971). Control of Uncertain Systems with a Set-Membership Description of Uncertainty. Cambridge, MA: PhD Thesis, MIT.
• Bertsekas, Dimitri P. (1999). Nonlinear Programming. Belmont, MA: Athena Scientific. pp. 737. ISBN 1-886529-00-0.