```markdown
## Steepest Descent Method

### Normalized Steepest Descent Direction
- **Formula**: \(\Delta x_{\text{nsd}} = \arg\min \{\nabla f(x)^T v \mid \|v\| = 1\}\)
- **Interpretation**: For small \(v\), \(f(x + v) \approx f(x) + \nabla f(x)^T v\).
- **Explanation**: Direction \(\Delta x_{\text{nsd}}\) is a unit-norm step with the most negative directional derivative.

### (Unnormalized) Steepest Descent Direction
- **Formula**: \(\Delta x_{\text{sd}} = \|\nabla f(x)\|_* \Delta x_{\text{nsd}}\)
- **Property**: Satisfies \(\nabla f(x)^T \Delta x_{\text{sd}} = -\|\nabla f(x)\|^2_*\).

### Steepest Descent Method
- **Description**: General descent method with \(\Delta x = \Delta x_{\text{sd}}\).
- **Properties**: Convergence properties similar to gradient descent.

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