Subdifferential sum rule for locally Lipschitz functions

Question

It is mentioned in this answer by @copper.hat that

As an aside, the notion of subdifferential can be extended to locally Lipschitz functions (subgradient) where the containment goes in the opposite direction (ignoring pathologies), that is the subgradients satisfy $\partial \sum_k f_k(x) \subset \sum_k \partial f_k(x)$.

Could you elaborate on a reference for this result?

Answer 1

I have found $2$ references.

1. Variational Analysis by Rockafellar/Wets.

9.1 Definition (Lipschitz continuity and strict continuity). Let $F$ be a singlevalued mapping defined on a set $D \subset \mathbb{R}^n$, with values in $\mathbb{R}^m$. Let $X \subset D$. The map $F$ is strictly continuous at $\bar{x}$ (without mention of $X$ ) if $\bar{x} \in \operatorname{int} D$ and the value $$ \operatorname{lip} F(\bar{x}):=\limsup _{\substack{x, x^{\prime} \rightarrow \bar{x} \\ x \neq x^{\prime}}} \frac{\left|F\left(x^{\prime}\right)-F(x)\right|}{\left|x^{\prime}-x\right|} $$ is finite.

9.17 Definition (strict differentiability). A function $f: \mathbb{R}^n \rightarrow \overline{\mathbb{R}}$ is strictly differentiable at a point $\bar{x}$ if $f(\bar{x})$ is finite and there is a vector $v$, which will be the gradient $\nabla f(\bar{x})$, such that $f\left(x^{\prime}\right)=f(x)+\left\langle v, x^{\prime}-x\right\rangle+o\left(\left|x^{\prime}-x\right|\right)$, i.e., $$ \frac{f\left(x^{\prime}\right)-f(x)-\left\langle v, x^{\prime}-x\right\rangle}{\left|x^{\prime}-x\right|} \rightarrow 0 \text { as } x, x^{\prime} \rightarrow \bar{x} \text { with } x^{\prime} \neq x . $$

10.10 Exercise (subgradients of Lipschitzian sums). If $f=f_1+f_2$ with $f_1$ strictly continuous at $\bar{x}$ while $f_2$ is lsc and proper with $f_2(\bar{x})$ finite, then $$ \partial f(\bar{x}) \subset \partial f_1(\bar{x})+\partial f_2(\bar{x}), \quad \partial^{\infty} f(\bar{x}) \subset \partial^{\infty} f_2(\bar{x}) . $$ If $f_1$ is strictly differentiable at $\bar{x}$, these inclusions hold as equations.

2. Optimization and Nonsmooth Analysis by Clarke.

We shall say that $F$ admits a strict derivative at $x$, an element of $\mathcal{L}(X, Y)$ denoted $D_s F(x)$, provided that for each $v$, the following holds: $$ \lim _{\substack{x^{\prime} \rightarrow x \\ t \downarrow 0}} \frac{F\left(x^{\prime}+t v\right)-F\left(x^{\prime}\right)}{t}=\left\langle D_s F(x), v\right\rangle, $$ and provided the convergence is uniform for $v$ in compact sets. (This last condition is automatic if $F$ is Lipschitz near $x$ ). Note that ours is a "Hadamard-type strict derivative".

If $f_i(i=1,2, \ldots, n)$ is a finite family of functions each of which is Lipschitz near $x$, it follows easily that their sum $f=\sum f_i$ is also Lipschitz near $x$.

2.3.3 Proposition (Finite Sums) $$ \partial\left(\sum f_i\right)(x) \subset \sum \partial f_i(x). $$ Corollary 1 Equality holds in Proposition 2.3.3 if all but at most one of the functions $f_t$ are strictly differentiable at $x$.