A theorem due to Danskin[1] allows one to study the regularity of function defined with an optimization problem.
Suppose $\phi (x,z)$ is a continuous function of two arguments, $\phi :{\mathbb {R} }^{n}\times Z\rightarrow {\mathbb {R} }$
where $Z\subset {\mathbb {R} }^{m}$ is a compact set. Further assume that $ \phi (x,z)$ is convex in $x$ for every $z\in Z$.
Define $f(x)=\max _{z\in Z}\phi (x,z)$ and the set of optimal solutions $Z_{0}(x)=\left\{{\overline {z}}:\phi (x,{\overline {z}})=\max _{z\in Z}\phi (x,z)\right\}$.
Danskin's theorem: $f(x)$ is differentiable at $x$ if $Z_{0}(x)$ consists of a single element $\overline {z}$. Furthermore, the derivative of $f(x)$ is given by $$ {\frac {\partial f}{\partial x}}={\frac {\partial \phi (x,{\overline {z}})}{\partial x}}.$$
I am looking for a variant of this theorem, where I would only have upper semicontinuity in the variable $z$ instead of continuity. I humbly think the extension is true as the continuity over the compact space is only used to claim the existence of a solution and thus to have a max instead of a sup. This argument remains true for upper semicontinuity. Do you know how any reference showing the Danskin theorem with this weaker asumption ?
I have found a variant from Clarke (theorem 2.1, [2]) that I could use, but it is really strong and not known as much as Dankin theorem.
[1]Danskin, John M. (1967). The Theory of Max-Min and its Applications to Weapons Allocation Problems. NY: Springer.
[2] CLARKE, FRANK H. (1975), GENERALIZED GRADIENTS AND APPLICATIONS, TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY