An extension of subgradients method for nonsmooth convex optimization

Question

Please excuse my ignorance on this question. Let us consider an extension of the problem here:

subdifferential rule proof

Suppose that $$f(x) = \max_{y\in Y} h(x, y)$$ where $Y$ is a compact set, and the functions $x \mapsto h(x,y)$ are continuous and convex for all $y \in Y.$

I am interested in the prove below but I hardly could prove this: $$\partial f(x) = \text{conv}\Big\{\bigcup \ \partial_x h(x,y) \ \Big| \ y \in Y \Big\}.$$

Answer 1

The result you ask about is an extension of Danskin's theorem. It is indeed true and was proved by Bertsekas in the appendix of his PhD thesis.