Directional derivative of proximal mapping of a convex function

Question

Let $f:\mathbb{R}^n\rightarrow\overline{\mathbb{R}}$ be a proper closed convex function that is locally Lipschitz continuous on its domain $D(f)$. Define the proximal mapping of $f$ to be $$\textbf{prox}_{\lambda f}(x)=\arg\min_u\left\{f(u)+\frac{1}{2\lambda}\|u-x\|^2\right\}$$ In this context, it is not hard to see $\textbf{prox}_{\lambda f}(x)$ is a Lipschitz continuous function w.r.t. variable $x$ for any parameter $\lambda>0$.

I am wondering under what assumption on $f$, will the directional derivative of proximal mapping, i.e., $$\textbf{prox}_{\lambda f}'(x;d) := \lim_{t\downarrow 0}\frac{\textbf{prox}_{\lambda f}(x+td)-\textbf{prox}_{\lambda f}(x)}{t}$$ exist?

From the existing literature (Proposition 5.3.5, p. 141), I know if $f:=I_C$ is an indicator function of a closed convex set $C$ (hence proximal mapping reduces to projection mapping), then the directional derivative exists. To be more precise, for any $x\in C$ and $d\in\mathbb{R}^n$, $$\textbf{proj}_C'(x;d) := \lim_{t\downarrow 0}\frac{\textbf{proj}_C(x+td)-\textbf{proj}_C(x)}{t} = \textbf{proj}_{T_C(x)}(d)$$ where $T_C(x)$ is the tangent cone of $C$ at $x$.

I am wondering if this is only special for projection operator or can be slightly generalized to proximal operator of convex functions with some nice properties or structures.

Answer 1

You must be careful, in your reference, the directional differentiability of the projection onto $C$ is only established for points $x \in C$. In fact, the projection might fail to be directionally differentiable at points outside $C$. For a counterexample, see the paper "Directionally nondifferentiable metric projection" by A. Shapiro.

For a general function $f$, one can prove that the directional differentiability of the prox is equivalent to $f$ being twice epi-differentiable. This should be in the book "Variational Analysis" by Rockafellar and Wets.

Indeed, one can argue as in Corollary 13.43 by using Theorem 13.40. Let $f$ be convex and twice epidifferentiable. Fix $u$, set $\newcommand\prox{\operatorname{prox}}x = \prox_f(u)$ and $v = u - x$. We have $$ \begin{split} \newcommand\dd{\mathrm{d}} \prox_f &= (I + \partial f)^{-1}\\ D(\partial f)(x,v) &= \partial [\frac12\dd^2f(x,v)]\\ D\prox_f(u,x) &= D(I + \partial f)^{-1}(u,x) = (D(I + \partial f)(x,u))^{-1} \\&= [DI(x,x) + D\partial f(x,u-x)]^{-1} \\ &=\partial [I + \partial\frac12\dd^2f(x,v)]^{-1} \\&= \prox_{\frac12 \dd^2 f(x,v)}. \end{split} $$ Hence, $z = \prox_f'(u; w) = D\prox_f(u,x)$ if $z$ minimizes $$ \frac12 \|z - w\|^2 + \frac12 \dd^2 f(x,v)(z). $$