the Banach fixed-point theorem and minimizer of a convex function

Question

I am not sure how to relate the Banach fixed-point theorem with the following proof:

say I have a function f(w) that maps w of shape(d, 1) to a scalar, let g(w) = w + α($\nabla$ f(w)); g is a contraction mapping and α is sufficiently small constant, probably with an upper bound. I also know that f is strictly convex, and has a minimizer $w^*$, and Banach fixed-point theorem tells us that g converges to its fixed point w0.

The overall purpose is to show w0 is actually the minimizer $w^*$ of f(w).

you can provide me with some hints or show the exact steps.

Answer 1

First of all, I am not sure what $shape(d,1)$ means Think about the following things:

1. What does it mean that $w^*$ is a minimizer of $f(w)$

2. What does it mean that the function is convex? Think about some necessary or sufficient conditions and think about the fact if there can be maximizers and if the minimizer is unique or are there more of them?

3. What does it mean that $w_0$ is a fixed point of $g(w)$

Write these things down an I would guess that you can see a bit more.