Danskin's theorem

In convex analysis, Danskin's theorem is a theorem which provides information about the derivatives of a function of the form $${\displaystyle f(x)=\max _{z\in Z}\phi (x,z).}$$

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.

The following version is proven in "Nonlinear programming" (1991). Suppose ${\displaystyle \phi (x,z)}$is a continuous function of two arguments, $${\displaystyle \phi :\mathbb {R} ^{n}\times Z\to \mathbb {R} }$$where ${\displaystyle Z\subset \mathbb {R} ^{m}}$is a compact set.

Under these conditions, Danskin's theorem provides conclusions regarding the convexity and differentiability of the function $${\displaystyle f(x)=\max _{z\in Z}\phi (x,z).}$$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 if ${\displaystyle \phi (x,z)}$is convex in ${\displaystyle x}$for every ${\displaystyle z\in Z}$.

Directional semi-differential

The semi-differential of ${\displaystyle f(x)}$in the direction ${\displaystyle y}$, denoted ${\displaystyle \partial _{y}\ f(x),}$is given by $${\displaystyle \partial _{y}f(x)=\max _{z\in Z_{0}(x)}\phi '(x,z;y),}$$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 $${\displaystyle {\frac {\partial f}{\partial x}}={\frac {\partial \phi (x,{\overline {z}})}{\partial x}}.}$$

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

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.

The 1971 Ph.D. Thesis by Dimitri P. Bertsekas (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 ${\displaystyle \operatorname {int} (\operatorname {dom} (f)),}$the interior of the effective domain of ${\displaystyle f,}$is nonempty, and that ${\displaystyle \phi }$is continuous on the set ${\displaystyle \operatorname {int} (\operatorname {dom} (f))\times Z.}$Then for all ${\displaystyle x}$in ${\displaystyle \operatorname {int} (\operatorname {dom} (f)),}$the subdifferential of ${\displaystyle f}$at ${\displaystyle x}$is given by $${\displaystyle \partial f(x)=\operatorname {conv} \left\{\partial \phi (x,z):z\in Z_{0}(x)\right\}}$$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
• Hotelling's lemma