```markdown
### Newton’s Method with Equality Constraints

**Procedure:**

- **Given:** Starting point \( x \in \text{dom} \, f \) with \( Ax = b \), tolerance \( \epsilon > 0 \).

- **Repeat:**
  1. Compute the Newton step and decrement \( \Delta x_{nt}, \lambda(x) \).
  2. **Stopping criterion:** Quit if \( \lambda^2/2 \leq \epsilon \).
  3. **Line search:** Choose step size \( t \) by backtracking line search.
  4. **Update:** \( x := x + t\Delta x_{nt} \).

**Properties:**

- A feasible descent method: \( x^{(k)} \) feasible and \( f(x^{(k+1)}) < f(x^{(k)}) \).
- Affine invariant.

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*Source: A. d’Aspremont. Convex Optimization M2. Slide 41/53.*
```