```markdown
## Newton Step

The slide describes the Newton step for a function \( f \) at a feasible point \( x \). The step is determined by solving the following system:

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

### Interpretations

- **\( \Delta x_{nt} \) Solves Second Order Approximation:**
  - The variable \( v \) is used in the approximation.
  - **Objective:**
    \[
    \text{minimize } \hat{f}(x + v) = f(x) + \nabla f(x)^T v + \frac{1}{2} v^T \nabla^2 f(x) v
    \]
  - **Subject to:**
    \[
    A(x + v) = b
    \]

- **Equations from Linearizing Optimality Conditions:**
  - The conditions are expressed as:
    \[
    \nabla f(x + \Delta x_{nt}) + A^T w = 0
    \]
    \[
    A(x + \Delta x_{nt}) = b
    \]

### Reference
- A. d'Aspremont, Convex Optimization M2, Slide 39/53
```
