Danskin's theorem

The following version is proven in "Nonlinear programming" (1991).^[2] 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 $${\displaystyle Z_{0}(x)=\arg \max _{z\in Z}\phi (x,z)=\left\{{\overline {z}}:\phi (x,{\overline {z}})=\max _{z\in Z}\phi (x,z)\right\}.}$$Danskin's theorem then provides the following results.

The 1971 Ph.D. Thesis by Dimitri P. Bertsekas (Proposition A.22) ^[3] 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