Let $a:R^k\to R$ and $b:R^k\to R$ be finite valued convex functions. Let $t_+=\max(0,t)$ be the positive part for any real $t\in R$. Define $$F(x)=\sqrt{a(x)^2_+ + b(x)^2_+}$$ I am looking for references (or a direct proof) regarding the subdifferential of the convex function $F$ at points $x$ such that $a(x)=0=b(x)$. The close result that comes to mind is Corollary 16.72 in the book Bauschke and P.L. Combettes: Convex Analysis and Monotone Operator Theory in Hilbert Spaces and results in that section of the book.
Remark: The subdifferential of the function $(a,b)\mapsto \sqrt{a_+^2+b_+^2}$ at $(0,0)$ is $$S = \{(u,v)\in R_{\ge 0}\times R_{\ge 0} : u^2+v^2\le 1\}.$$
Questions:
- What is the subdifferential of $F$ when $a$ and $b$ are differentialbe at $x$? Is it provably $\cup_{(u,v)\in S} (u\nabla a(x) + v\nabla b(x) )$?
- What is the subdifferential of $F$ without differentiability of $a,b$? Is it always $\cup_{(u,v)\in S} (u\partial a(x) + v\partial b(x))$?
Note that in both cases, the inclusions $u\nabla a(x) + v\nabla b(x)\subset \partial F(x)$ and $u\partial a(x) + v\partial b(x)\subset \partial F(x)$ are straightfoward. The question is whether equality holds.