Suppose that f : X → ]−∞, +∞] is convex and proper, that x ̄ ∈ dom f and that λ > 0. Show that

Question

Suppose that $f : X → ]−\infty, +\infty]$ is convex and proper, that $\bar{x} \in dom f$, and that $\lambda > 0$. Show that

$$\partial^{\infty}(\lambda f)(\bar{x}) = \lambda\partial^{\infty}f(\bar{x}).$$

I have this property but I am still unsure where to start with the proof.

$$v\in \partial f(\bar{x})\iff(\forall\;x\in X)f(\bar{x})\langle v,x-\bar{x}\rangle\leq f(x). $$

Answer 1

For this type of problem you typically want to choose some element you know to exist on the left hand side and algebraically show it must also be contained on the right hand side. For this problem specifically we want to chose $v\in\partial^{\infty}(\lambda f)(\bar{x})$ and conclude that $v\in\lambda\partial^{\infty}f(\bar{x}).$ This can be done entirely with iff statements. Also, it will be useful to recall that singular sub differential at a point x is equal to the normal cone over the domain of f at x. That is to say $$\partial^{\infty}f(\bar{x})=N_{dom(f)}(\bar{x})$$ where the normal cone is defined as $$N_{dom(f)}(\bar{x})= \{v\in\mathbb{R}:\langle v,x-\bar{x}\rangle\leq0\;\forall x\in dom(f)\}.$$

Now that we have all the required information I will get you started with the actual proof:

Observe that \begin{align} v\in\partial^{\infty}(\lambda f)(\bar{x})&\iff v\in N_{dom(\lambda f)}(\bar{x})\\ &\iff \langle (v,0),(x,( f)(\lambda x))-(x,( f)(\lambda \bar{x})) \rangle\leq0\\ &\quad\vdots\\ &\iff v\in\lambda\partial^{\infty}f(\bar{x}). \end{align}