```markdown
### Infeasible Start Newton Method

#### Algorithm Description:
- **Given:** Starting point \( x \in \text{dom } f \), \( \nu \), tolerance \( \epsilon > 0 \), \( \alpha \in (0, 1/2) \), \( \beta \in (0, 1) \).
- **Repeat:**
  1. Compute primal and dual Newton steps \( \Delta x_{nt}, \Delta \nu_{nt} \).
  2. Perform backtracking line search on \( \| r \|_2 \):
     - Set \( t := 1 \).
     - While \( \| r(x + t \Delta x_{nt}, \nu + t \Delta \nu_{nt}) \|_2 > (1 - \alpha t) \| r(x, \nu) \|_2 \), set \( t := \beta t \).
  3. Update: \( x := x + t \Delta x_{nt}, \nu := \nu + t \Delta \nu_{nt} \).
- **Until:** \( Ax = b \) and \( \| r(x, \nu) \|_2 \leq \epsilon \).

#### Notes:
- Not a descent method: \( f(x^{(k+1)}) > f(x^{(k)}) \) is possible.
- Directional derivative of \( \| r(y) \|_2^2 \) in direction \( \Delta y = (\Delta x_{nt}, \Delta \nu_{nt}) \) is:
  \[
  \frac{d}{dt} \left\| r(y + \Delta y) \right\|_2 \bigg|_{t=0} = -\| r(y) \|_2
  \]

#### Reference:
- A. d’Aspremont. Convex Optimization M2. Slide 44/53.
```
