Created: August 13, 2022
Modified: August 13, 2022

Danskin's theorem

This page is from my personal notes, and has not been specifically reviewed for public consumption. It might be incomplete, wrong, outdated, or stupid. Caveat lector.

A pointwise maximum of convex functionsSpecifically, we require that $\phi(x, z)$ is convex in $x$ for every $z$ .

f(x) = \max_z \phi(x, z),

is itself convex in $x$ , and when the maximizing point

\bar{z}(x) = \arg\max_z \phi(x, z)

is unique then $f$ is differentiable at $x$ with derivative

\frac{d}{dx} f(x) = \frac{\partial \phi(x, \bar{z}(x))}{\partial x}.

This immediately provides a gradient for solutions to convex optimization problems.

Danskin's theorem

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