```markdown
### Line Search Types

#### Exact Line Search
- Formula: \( t = \arg\min_{t > 0} f(x + t \Delta x) \)

#### Backtracking Line Search
- Parameters: \( \alpha \in (0, 1/2) \), \( \beta \in (0, 1) \)
- Procedure:
  - Start at \( t = 1 \)
  - Repeat: \( t := \beta t \) until
  \[
  f(x + t \Delta x) < f(x) + \alpha t \nabla f(x)^T \Delta x
  \]

#### Graphical Interpretation
- Backtrack until \( t \leq t_0 \)

#### Graph
- A plot is shown with:
  - A curve representing \( f(x + t \Delta x) \)
  - Two dashed lines:
    - One for \( f(x) + t \nabla f(x)^T \Delta x \)
    - One for \( f(x) + \alpha t \nabla f(x)^T \Delta x \)
  - Points labeled at \( t = 0 \) and \( t = t_0 \)

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*A. d'Aspremont. Convex Optimization M2.*
```
