```markdown
### Newton's Method

**Algorithm:**

- **Given** a starting point \( x \in \text{dom } f \), tolerance \( \epsilon > 0 \).
- **Repeat:**
  1. **Compute the Newton step and decrement.**
     \[
     \Delta x_{\text{nt}} := -\nabla^2 f(x)^{-1} \nabla f(x);
     \]
     \[
     \lambda^2 := \nabla f(x)^T \nabla^2 f(x)^{-1} \nabla f(x).
     \]
  2. **Stopping criterion.** Quit if \( \lambda^2 / 2 \leq \epsilon \).
  3. **Line search.** Choose step size \( t \) by backtracking line search.
  4. **Update.** \( x := x + t \Delta x_{\text{nt}} \).

- **Affine invariant**, i.e., independent of linear changes of coordinates:

Newton iterates for \( \tilde{f}(y) = f(Ty) \) with starting point \( y^{(0)} = T^{-1}x^{(0)} \) are
\[ 
y^{(k)} = T^{-1}x^{(k)} 
\]

---

**Source:** A. d’Aspremont. Convex Optimization M2. Slide 19/53.
```
