Has research been conducted on the topic of directional derivatives of functions that are minimums of convex functions? It would be greatly appreciated if you could provide the appropriate reference.
To be specific, say $z^k:\mathbb{R}^n\rightarrow\mathbb{R}\cup{+\infty} $ is a given convex function for each $k=1,\dots,K$. And, let $z:\mathbb{R}^n\rightarrow\mathbb{R}\cup{+\infty} $ be defined as follows: $z(x) = \min_{k=1,\dots,K} z^k(x)$. Intuitively, the directional derivatives of the function $z$ at point $x$ in direction $d$ should match the directional derivatives of the function $z^k $ at point $x$ in direction $d$, where the function $z$ and function $z^k $ coincide at point $x$ (or where the function $z^k $ is active at point $x$). I am seeking a formal proof of this concept in the literature.
z^k:\mathbb{R}^n\rightarrow\mathbb{R}\cup{+\infty}is a given convex function for eachk=1,\dots,K.And, letz:\mathbb{R}^n\rightarrow\mathbb{R}\cup{+\infty}be defined as follows:z(x) = \min_{k=1,\dots,K} z^k(x).Intuitively, the directional derivatives of the function $z$ at point $x$ in direction $d$ should match the directional derivatives of the function $z^k$ at point $x$ in direction $d$, where the function $z$ and function $z^k$ coincide at point $x$ (or where the function $z^k$ is active at point $x$). I am seeking a formal proof of this concept in the literature. $\endgroup$