Gradient of Perspective function

Question

I was reading the convex optimization book from Prof Boyd and Prof Vandenberghe and In chapter 3, I had a doubt regarding the perspective function.

In page 89, the perspective function of a function f is defined as :

$g(x,t)=tf(x/t)$

assuming $f: R^{n}→R$ so that $g:R^{n+1}→R$ and also $x,t∈ dom(f),t>0$

I was wondering what would be the gradient of $g(x,t)$ in terms of $t$ and $f(x/t)$. Please, can someone help me with this? Thank you.

Answer 1

The gradient of $g$ follows from chain rule:

$$\nabla_{n+1} g(x,t) = \left( \nabla_n f\left(\frac{x}{t}\right),\: f\left(\frac{x}{t}\right) - \frac{x}{t}\cdot\nabla_n f\left(\frac{x}{t}\right) \right)$$

where the "first" term is really a list of entries of length $n$.