# 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 $$\Psi(x):X\rightarrow \mathbb{R}$$ 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

• 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. Commented Aug 3, 2016 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? Commented Dec 3, 2014 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.