```markdown
### Newton Step at Infeasible Points

**Context:**
- The 2nd interpretation from page 39 extends to infeasible \( x \) (i.e., \( Ax \neq b \)).
- Linearizing optimality conditions at infeasible \( x \) (with \( x \in \text{dom } f \)) gives:

\[
\begin{bmatrix}
\nabla^2 f(x) & A^T \\
A & 0 
\end{bmatrix}
\begin{bmatrix}
\Delta x_{nt} \\
w 
\end{bmatrix}
= - 
\begin{bmatrix}
\nabla f(x) \\
Ax - b 
\end{bmatrix}
\]

**Primal-Dual Interpretation:**
- Write the optimality condition as \( r(y) = 0 \), where:
  - \( y = (x, \nu) \)
  - \( r(y) = (\nabla f(x) + A^T \nu, Ax - b) \)

- Linearizing \( r(y) = 0 \) gives \( r(y + \Delta y) \approx r(y) + Dr(y)\Delta y = 0 \):

\[
\begin{bmatrix}
\nabla^2 f(x) & A^T \\
A & 0 
\end{bmatrix}
\begin{bmatrix}
\Delta x_{nt} \\
\Delta \nu_{nt} 
\end{bmatrix}
= - 
\begin{bmatrix}
\nabla f(x) + A^T \nu \\
Ax - b 
\end{bmatrix}
\]

- Same as (1) with \( w = \nu + \Delta \nu_{nt} \).

**Source:**
- A. d'Aspremont, Convex Optimization M2, Slide 43/53.
```
