Is there Lipschitz property for subdifferential?

I'm trying to bound the quantity $\langle \nabla \Psi(x),\bar{x}-x \rangle$ above, with the bound depending on $\|x-\bar{x}\|$ and perhaps also of $\|x-y\|$ for fixed (but not varying) points $y$. Where here $\Psi(x):X\mapsto \mathbb{R}$ with $X$ a finite dimensional Banach space (or $\mathbb{R}^n$, whatever) And $\Psi(x)$ is a $\mu$-strongly convex function (with $\mu$>0) that can be written as $\Psi=f+g$ with $f$ convex and differentiable with $\nabla f$ $L$-Lipschitz continuous and $g$ $\mu$-strongly convex.

I know that if $\Psi$ was differentiable and its gradient was $L$-Lipschitz continuous one could fix some point $x^*$ on the optimal set and bound as

$\langle \nabla \Psi(x), \bar{x}-x \rangle \leq \|\nabla \Psi(x)\|\|\bar{x}-x\| = \|\nabla \Psi(x)-\nabla \Psi(x^*)\|\|\bar{x}-x\| \leq L\|x-x^*\|\|\bar{x}-x\|$

And the bound is done. So my question is, is there an analogous of this property on the non-differentiable case? Like, I know that I can pick a point $x^*$ on the optimal set such that $0 \in \partial \Psi(x^*)$, but then can I say that for a $v \in \partial \Psi(x)$ it holds

$\|v\| = \|v-0\| \leq L\|x-x^*\|$ or something on that line?

Any help is appreciated

• Oops, yes I am. Fixed I guess – karlabos Dec 2 '14 at 18:28
• There is a concept of Lipschitz continuity for set-valued mappings as discussed in Rocafellar and Wets, Variational Analysis, Sec 9-E and, this said, some subgradients can be found to be Lipschitz continuous. – Pantelis Sopasakis Aug 3 '16 at 8:41

The answer is no. On the real line consider $\Phi(x)=|x|$ (and add some smooth convex function with minimum in zero if you like). Then the minimum is in zero but the subgradient at any positive point is about 1.
• But then, for any points $x,\bar{x}$, for every $v \in \partial \Phi(x)$ we'd have $\langle \nabla \Phi(x), x-\bar{x} \rangle \leq \|\nabla \Phi(x)\|\|x-\bar{x}\| \leq 1\|x-\bar{x}\|$, so we have the bound... Am I missing something on this counterexample? – karlabos Dec 3 '14 at 0:08
I think the property you are looking for is Lipschtiz continuity of the function $\Psi$, as this is equivalent to bounded subgradients.