```markdown
### Newton's Method and Elimination

**Newton's method for reduced problem**

- Minimize \(\tilde{f}(z) = f(Fz + \hat{x})\)
- **Variables:** \(z \in \mathbb{R}^{n-p}\)
- \(\hat{x}\) satisfies \(A\hat{x} = b\); Rank \(F = n - p\) and \(AF = 0\)
- Newton’s method for \(\tilde{f}\), started at \(z^{(0)}\), generates iterates \(z^{(k)}\)

**Newton's method with equality constraints**

- When started at \(x^{(0)} = Fz^{(0)} + \hat{x}\), iterates are \(x^{(k+1)} = Fz^{(k)} + \hat{x}\)
- Hence, don’t need separate convergence analysis

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*Source: A. d’Aspremont. Convex Optimization M2. Slide 42/53*
```